Trigonometric Identities(Math | Trig | Identities)

sin(theta) = a / c csc(theta) = 1 / sin(theta) = c / a

cos(theta) = b / c sec(theta) = 1 / cos(theta) = c / b

tan(theta) = sin(theta) / cos(theta) = a / b cot(theta) = 1/ tan(theta) = b / a

sin(-x) = -sin(x)csc(-x) = -csc(x)cos(-x) = cos(x)sec(-x) = sec(x)tan(-x) = -tan(x)cot(-x) = -cot(x)sin^2(x) + cos^2(x) = 1 tan^2(x) + 1 = sec^2(x) cot^2(x) + 1 = csc^2(x)

sin(x  y) = sin x cos y  cos x sin y

cos(x  y) = cos x cosy  sin x sin y

tan(x  y) = (tan x  tan y) / (1   tan x tan y)

sin(2x) = 2 sin x cos x

cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)

tan(2x) = 2 tan(x) / (1 - tan^2(x))

sin^2(x) = 1/2 - 1/2 cos(2x)

cos^2(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )

Trig Table of Common Anglesangle 0 30 45 60 90

sin^2(a) 0/4 1/4 2/4 3/4 4/4cos^2(a) 4/4 3/4 2/4 1/4 0/4

tan^2(a) 0/4 1/3 2/2 3/1 4/0

Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:

a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)

c^2 = a^2 + b^2 - 2ab cos(C)

b^2 = a^2 + c^2 - 2ac cos(B)

a^2 = b^2 + c^2 - 2bc cos(A)

(Law of Cosines)

(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)

sin(theta) = a / c

csc(theta) = 1 / sin(theta) = c / a

cos(theta) = b / c

sec(theta) = 1 / cos(theta) = c / b

tan(theta) = sin(theta) / cos(theta) = a / b

cot(theta) = 1/ tan(theta) = b / a

sin(-x) = -sin(x)

csc(-x) = csc(x)

cos(-x) = cos(x)

sec(-x) = sec(x)

tan(-x) = -tan(x)

cot(-x) = -cot(x)

sin^2(x) + cos^2(x) = 1

tan^2(x) + 1 = sec^2(x)

cot^2(x) + 1 = csc^2(x)

sin(x y) = sin x cos y cos x sin y

cos(x y) = cos x cos y sin x sin y

tan(x y) = (tan x tan y) / (1  tan x tan y)

sin(2x) = 2 sin x cos x cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)

tan(2x) = 2 tan(x) / (1 - tan^2(x))

sin^2(x) = 1/2 - 1/2 cos(2x)

cos^2(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )

cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )

Trig Table Of Common Angles

Angle 0 30 45 60 90

sin^2(a) 0/4 1/4 2/4 3/4 4/4

cos^2(a) 4/4 3/4 2/4 1/4 0/4

tan^2(a) 0/4 1/3 2/2 3/1 4/0

Sides a,b,c - Angles A,B,Ca Opposite A, b Opposite B, c Opposite C

Law Of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Law Of Cosines

c^2 = a^2 + b^2 - 2ab cos(C)

b^2 = a^2 + c^2 - 2ac cos(B)

a^2 = b^2 + c^2 - 2bc cos(A)

Law Of Tangents

(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2]

Basic angles

Since the trigonometric ratios for most angles cannot be calculated exactly in closed algebraic form, a few well-known angles that can be calculated often comprise the bulk of textbook exercises involving trigonometry.

The basic angles are given in Table 1.

Table 1: Basic angles encountered in trigonometry

sin cos tan

0 0 1 0

30 1/2 3 2  1 3 

45 2 2  2 2  1

60 3 2  1/2 3 

90 1 0

Other angles by addition and halving

These basic angles can be easily extended to obtain more angles of interest. Adding multiples of 90  merely rotates these angles into other quadrants; the appropriate values ofsin and cos can be obtained through symmetry.The values for 15  can be obtained by using the formula for the difference of angles:

   

     

     

     

Likewise, we can find that

   

   

   

More exact angles can be obtained by solving the double angle identity:

sin2 = 21−cos cos2 = 21+cos  

So for example, sin7 5 = (4− 6− 2) 8  . These angles can be further added and subdivided to obtain a dense subset of exactly known angles. However, such effort is not generally useful. Computers and calculators use a combination of lookup-tables and numeric iteration to obtain their values.

The angles 18   , 36   , 54   , 72  The 18  -36  -54  -72  series of angles cannot be obtained by halving, doubling, adding or subtracting the previous angles. Nevertheless, they are constructible, and their exact values can be derived by the following elementary procedure:Consider an isosceles triangle with the angles 72  , 54  and 54  . From the triangle we derive the relation:

sin272 =cos54

Notice that 72=4 18 and 54=3 18 , so if x=18 , then

   

   

   

   

The last equation is a quadratic equation that can be solved for sin18  . Carrying out the calculations, we obtain the values in Table 2.

Table 2: Other constructible angles in trigonometry

sin cos

18 4 5−1  2 2 5+ 5 

36 2 2 5− 5  4 5+1 

54 4 5+1  2 2 5− 5 

72 2 2 5+ 5  4 5−1 

First, you need a protractor. Create a right triangle ABC with C as the right triangle. Let angle B be your angle. Make rough measurements of the three sides of the triangle. now:sin B = AC/AB

Once you know its sine, use these formulae for the other trigonometric functions:cos B = sqrt(1 - sin² B)tan B = sin B/cos B

Special cases are:sin 0 = 0sin 90 = 1sin 30 = 1/2sin 60 = sqrt3/2sin 45 = sqrt2/2From these, you can compute all angles.remember,sin B = sin (B - 360n) where n is an integersin B = cos (90 - B)cos B = sin (90 - B)cos (-B) = cos Bsin (-B) = -sin B

example:

calculate sin, cos and tan of -150.first, sin -150 = -sin 150-sin 150 = -cos(90 - 150)= -cos (-60)= - cos 60 = -sin(90 - 60) = - sin 30= -1/2sin -150 = -1/2cos -150 = cos 150 = sin(90 - 150) = sin (-60) = -sin 60= -sqrt3/2tan -150 = sin -150/cos -150= (-1/2)/(-sqrt3/2)= sqrt3/3

Just use these sign rules:if angle is in Q1 = sin cos and tan are +if angle is in Q2 = sin is +,cos is -,tan is -if angle is in Q3 = sin is -,cos is -,tan is +is angle is in Q4 = sin is -,cos is +,tan is –

TRIGONOMETRY(i)  

 Contents:

Right Angled Triangles: Formulae:

One Angle & One Side Examples:

Two Sides & No Angle Examples:

Applied Examples: Obtuse Angles:

 

Right Angled Triangles: Trigonometry equations interconnect the angles and sides of right angled triangles.

The longest side is always called the hypotenuse and is always opposite the right angle. The other two sides of the triangle are named in relation to the subject angle. The first triangle in the diagram above shows the subject angle in blue. These other sides make up the right angle and are named as opposite to the subject angle, and adjacent to the subject angle. The second triangle above shows the names of the sides when the subject angle is changed.

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Formulae: where   is the subject angle:

The ratios between the sides are functions of the subject angle.

sin   = opposite hypotenuse.

cos   = adjacent hypotenuse.

tan   = opposite adjacent.

These are useful where an angle and the length of one side is given, then all the other lengths can be found.

 = sin  opposite hypotenuse.

 = cos  adjacent hypotenuse.

 = tan  opposite adjacent.

(The   means the inverse, usually a shift key on the calculator). These are useful where the lenght of at least two sides are known, but no angles given (other than the right angle).

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One Angle & One Side Examples: (A calculator is required to follow these examples). The names of the sides are given after the subject angle is known. Here the subject angle is drawn in blue in the diagram below.

opp = opposite, adj = adjacent, hyp = hypotenuse

(i) One angle and the hypotenuse is given, find the other two sides:

Formula: sin 27  = opp hyp. Rearrange: opp = sin 27 x hyp. Therefore opp = 36.

Formula: cos 27  = adj hyp. Rearrange: adj = cos 27  x hyp. Therefore adj = 71.

(ii) One angle and the opposite side is given, find the other two sides:

Formula: sin 27  = opp hyp. Rearrange: hyp = opp sin 27 . Therefore hyp = 132.

Formula: tan 27  = opp adj. Rearrange: adj = opp tan 27 . Therefore adj = 118.

(iii) One angle and the adjacent side is given, find the other two sides:

Formula: cos 27  = adj hyp. Rearrange: hyp = adj cos 27 . Therefore hyp = 79.

Formula: tan 27  = opp adj. Rearrange: opp = adj x tan 27 . Therefore opp = 36.

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Exercise (a):

Exercise (b):

Top of Page Back to One Angle One Side Examples  Back to Formulae

Two Sides & No Angles Examples: (A calculator is required to follow these examples). The names of the sides are given with respect to the required angle. Here the required angle is drawn in blue in the diagram below.

opp = opposite, adj = adjacent, hyp = hypotenuse.

(iv) Two sides are given, find the angle 'a':

The known sides are opposite and hypotenuse, therefore use sin function. Formula: a  = sin  x opp hyp. = 30 .

(The other angle is therefore 60  (all angles sum to 180 ). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths are now the hypotenuse and the adjacent sides. Formula: angle  = cos x adj hyp. = 60 .)

(v) Two sides are given, find the angle 'b':

The known sides are the adjacent and the hypotenuse, therefore use the cos function. Formula: b  = cos  x adj hyp. = 389 .

(The other angle is therefore 511 (all angles sum to 180 ). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths are now the hypotenuse and the opposite sides. Formula: angle  = sin x opp hyp. = 511 .)

(vi) Two sides are given, find the angle 'c':

The known sides are the opposite and the adjacent, therefore use the tan function. Formula: c  = tan  x opp adj. = 575 .

(The other angle is therefore 325 (all angles sum to 180 ). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths still the opposite and the adjacent sides but reversed. Formula: angle  = tan  x opp adj. = 325 .)

Top of Page  Back to Two Sides & No Angles Examples

Exercise (c):

Exercise (d):

Top of Page  Back to Two Sides & No Angles Examples  Back to Formulae

Applied Examples:

(vii)

A church tower stands 35 metres high, its shadow is measured at 50 metres. Find the angle of elevation of the sun ?

 

 

Answer: Draw the right angled triangle: Use the formula where two sides are known but no angle: From the angle in question the two sides known are the opposite and the adjacent. Therefore the formula is:   = tan  x 35 50 = 35 .

(viii) A plane is flying at 10,000 metres altitude directly above, 12 seconds later the angle of elevation is 75 , find the speed of the plane.

Answer: Draw the right angled triangle: Use the formula where one side is known and one angle is known: The side that is needed is the side A B. Using rules from Geometry(i) the angle within the triangle next to the angle of elevation is 15  also the angle at B is 75 . Using this angle at Bthe formula is tan 75  = 10000 adj. Rearranging adj or A B = 10000 tan 75  = 2680 metres. The speed is 2680 metres in 12 seconds or (x 300) 804000 metres in an hour ( 804 kmh).

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Obtuse Angles:

 

The trigonometric ratios of sin, cos and tan are defined as follows:

sin (180  -  ) = opp hyp.

-cos (180  -  ) = adj hyp.

-tan (180  -  ) = opp adj.

Example: if   = 160 ; sin = sin 20 ; cos = - cos 20 ; tan = - tan 20 .

Example: find the obtuse angle when sin   = 06.

Answer: sin  06 = 369. Therefore: 180  - 369  = 1431 .

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©Mathstutor.com  2001-2005

Sine, Cosine and TangentThree Functions, but same idea.

Right TriangleSine, Cosine and Tangent are all based on a Right-Angled Triangle

Before getting stuck into the functions, it helps to give a name to each side of a right triangle: 

"Opposite" is opposite to the angle θ

"Adjacent" is adjacent (next to) to the angle θ

"Hypotenuse" is the long one

Adjacent is always next to the angle

And Opposite is opposite the angle

Sine, Cosine and TangentSine, Cosine and Tangent are the three main functions in trigonometry.

They are often shortened to sin, cos and tan.

To calculate them:

Divide the length of one side by another side ... but you must know which sides!

For a triangle with an angle θ, the functions are calculated this way:

Sine Function: sin(θ) = Opposite / Hypotenuse

Cosine Function: cos(θ) = Adjacent / Hypotenuse

Tangent Function: tan(θ) = Opposite / Adjacent

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):sin(35°) = Opposite / Hypotenuse = 2.8 / 4.9 = 0.57...

 Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.

But you still need to remember what they mean!

"Why didn't sin and tan go to the party?" "... just cos!"

ExamplesExample: what are the sine, cosine and tangent of 30° ?The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √(3):

Now we know the lengths, we can calculate the functions:Sine sin(30°) = 1 / 2 = 0.5

Cosine cos(30°) = 1.732 / 2 = 0.866...

Tangent tan(30°) = 1 / 1.732 = 0.577...

(get your calculator out and check them!)

 

Example: what are the sine, cosine and tangent of 45° ?The classic 45° triangle has two sides of 1 and a hypotenuse of √(2):

Sine sin(45°) = 1 / 1.414 = 0.707...

Cosine cos(45°) = 1 / 1.414 = 0.707...

Tangent tan(45°) = 1 / 1 = 1

View Larger

Try It!Have a try! Drag the corner around to see how different angles (inradians or degrees) affect sine, cosine and tangent.

In this animation the hypotenuse is 1, making the Unit Circle.

Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also.

 

 

SohcahtoaSohca...what? Just an easy way to remember which side to divide by which! Like this:

Soh... Sine = Opposite / Hypotenuse

...cah... Cosine = Adjacent / Hypotenuse

...toa Tangent = Opposite / Adjacent

You can read more about sohcahtoa ...

... but please remember "sohcahtoa" - it could help in an exam !

 

Why?Why are these functions important?

Because they let you work out angles when you know sides And they let you work out sides when you know angles

 Example: Use the sine function to find "d"We know* The angle the cable makes with the seabed is 39°* The cable's length is 30 m.And we want to know "d" (the distance down).  

Start with: sin 39° = opposite/hypotenuse = d/30

Swap Sides: d/30 = sin 39°

Use a calculator to find sin 39°: d/30 = 0.6293…

Multiply both sides by 30: d = 0.6293… x 30 = 18.88 to 2 decimal places.

The depth "d" is 18.88 m

 

ExerciseTry this paper-based exercise where you can calculate the sine function for all angles from 0° to 360°, and then graph the result. It will help you to understand these relatively simple functions.

You can also see Graphs of Sine, Cosine and Tangent.

Less Common FunctionsTo complete the picture, there are 3 other functions where you divide one side by another, but they are not so commonly used.

They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:

Secant Function: sec(θ) = Hypotenuse / Adjacent (=1/cos)

Cosecant Function: csc(θ) = Hypotenuse / Opposite (=1/sin)

Cotangent Function: cot(θ) = Adjacent / Opposite (=1/tan)

Formulas for Sine and Cosine

Date: 1/24/96 at 21:23:51Subject: What are the definitions of sin, cos, and tan in terms of theta?From: Michael Allman

Dr. Math,

I have been searching for an answer for several days and am really stuck. Even my precalc. teacher doesn't know (by the way - I am a high school junior.) What are the definitions of sin, cos, and tan in terms of theta? So far, I have come upon a definition that looks something like:

sin x = x + x^3/3! - x^5/5! + x^7/7! . . .

but I do not know how to handle this series. Also, because of the ambiguous use of the variable x, I do not even know if this series is the answer I'm looking for. So, is there a nice neat answer?

I also have a question about the notation of the trig. functions. If sin is a function and f(x) describes a function of x (e.g. f(x) = 2x), why is sin theta defined by y and r in my textbook? It seems to me that sin theta should be defined by theta and that sin (y,r) should be defined by y and r.

In a nutshell:

1. What is the definition of sin, cos, and tan in terms of the angle (not in terms of x, y, or r?)

2. Why is the function sin theta defined by y and r, and not by theta?

Thank you for your help!

Michael Allman

Date: 1/27/96 at 17:3:42From: Doctor KenSubject: Re: What are the definitions of sin, cos, and tan in terms...

Hello!

The way Sine and Cosine are defined is usually in terms of the unit circle. What you do is draw a circle of radius 1, whose center is at the point (0,0). Then you draw a ray coming out from the origin that makes an angle of theta with the x-axis. Note that the way you measure this angle is by starting at the x-axis, and travelling COUNTER-CLOCKWISE until you hit the ray in question. Thus this is about 60 degrees: / / / / / theta/_________________> x-axis

And this is about -60 degrees:

__________________> x-axis\ theta \ \ \ \ \

This ray will intersect with the unit circle at a point (x,y). Cos(theta) is defined as x, the first coordinate of this intersection point, and Sin(theta) is defined as y, the second coordinate.

Then Tangent is defined as Sin/Cos, and so on.

>2. Why is the function sin theta defined by y and r, and not by theta?

Well, I'm not really sure what you mean by y and r. What are your y and r?

-Doctor Ken, The Geometry Forum

Date: 1/29/96 at 15:1:2Subject: Re: What are the definitions of sin, cos, and tan in terms of From: Michael Allman

Dr. Math,

You misunderstood my two questions. The first one asked for thedefinitions of the trigonometric functions in terms of theta, theta being the angle in question. So if I needed to know the value of sin 54.2 degrees and I did not have a calculator, how would I calculate this? When you wrote "Cos(theta) is defined as x, the first coordinate of this intersection point, and Sin(theta) is defined as y,...", you defined cos and sin in terms of the coordinates x and y.

As for my second question, y stands for the y coordinate of a point in a plane, and r stands for the length of the segment between that same point and the origin of the coordinate system. See below.

y-axis (x,y) | / | / | r / | / | / | / | / | / _____|/______________________ x-axis origin | | |

So, since sin theta is a function of theta, why is it defined by (y/r) and not by theta itself? This seems to be an incongruity in function notation.

I hope this is clearer. Thank you.

An Introduction to Trigonometry ... by Brandon WilliamsMain Index...

Introduction Well it is nearly one in the morning and I have tons of work to do and a fabulous idea pops into my head: How about writing an introductory tutorial to trigonometry! I am going to fall so far behind. And once again I did not have the chance to proof read this or check my work so if you find any mistakes e-mail me.

I'm going to try my best to write this as if the reader has no previous knowledge of math (outside of some basic Algebra at least) and I'll do my best to keep it consistent. There may be flaws or gaps in my logic at which point you can e-mail me and I will do my best to go back over something more specific. So let's begin with a standard definition of trigonometry:

trig - o - nom - e - try n. - a branch of mathematics which deals with relations between sides and angles of triangles 

Basics Well that may not sound very interesting at the moment but trigonometry is the most interesting forms of math I have come across…and just to let you know I do not have an extensive background in math. Well since trigonometry has a lot to do with angles and triangles let's familiarize ourselves with some fundamentals. First a right triangle: 

 

A right triangle is a triangle that has one 90-degree angle. The 90-degree angle is denoted with a little square drawn in the corner. The two sides that are adjacent to the 90-degree angle, 'a' and 'b', are called the legs. The longer side opposite of the 90-degree angle, 'c', is called the hypotenuse. The hypotenuse is always longer than the legs. While we are on the subject lets brush up on the Pythagorean Theorem. The Pythagorean Theorem states that the sum of the two legs squared is equal to the hypotenuse squared. An equation you can use is:

c^2 = a^2 + b^2

So lets say we knew that 'a' equaled 3 and 'b' equaled 4 how would we find the length of 'c'…assuming this is in fact a right triangle. Plug-in the values that you know into your formula:

c^2 = 3^2 + 4^2

Three squared plus four squared is twenty-five so we now have this:

c^2 = 25 - - - > Take the square root of both sides and you now know that c = 5

So now we are passed some of the relatively boring parts. Let's talk about certain types of right triangles. There is the 45-45-90 triangle and the 30-60-90 triangle. We might as well learn these because we'll need them later when we get to the unit circle. Look at this picture and observe a few of the things going on for a 45-45-90 triangle:

In a 45-45-90 triangle you have a 90-degree angle and two 45-degree angles (duh) but also the two legs are equal. Also if you know the value of 'c' then the legs are simply 'c' multiplied by the square root of two divided by two. I rather not explain that because I would have to draw more pictures…hopefully you will be able to prove it through your own understanding. The 30-60-90 triangle is a little but harder to get but I am not going into to detail with it…here is a picture:

 

You now have one 30-degree angle, a 60-degree angle, and a 90-degree angle. This time the relationship between the sides is a little different. The shorter side is half of the hypotenuse. The longer side is the hypotenuse times the square root of 3 all divided by two. That's all I'm really going to say on this subject but make sure you get this before you go on because it is crucial in understanding the unit circle…which in turn is crucial for understanding trigonometry. 

Trigonometric Functions The entire subject of trigonometry is mostly based on these functions we are about to learn. The three basic ones are sine, cosine, and tangent. First to clear up any confusion that some might have: these functions mean nothing with out a number with them i.e. sin (20) is something…sin is nothing. Make sure you know that. Now for some quick definitions (these are my own definitions…if you do not get what I am saying look them up on some other website): 

Sine - the ratio of the side opposite of an angle in a right triangle over the hypotenuse. 

Cosine - the ratio of the side adjacent of an angle in a right triangle over the hypotenuse. 

Tangent - the ratio of the side opposite of an angle in a right triangle over the adjacent side.

Now before I go on I should also say that those functions only find ratios and nothing more. It may seem kind of useless now but they are very powerful functions. Also I am only going to explain the things that I think are useful in Flash…I could go off on some tangent (no pun intended) on other areas of Trigonometry but I'll try to keep it just to the useful stuff. OK lets look at a few pictures:

 

Angles are usually denoted with capital case letters so that is what I used. Now lets find all of the trigonometry ratios for angle A:

sin A = 4/5 cos A = 3/5 tan A = 4/3 

Now it would be hard for me to explain more than what I have done, for this at least, so you are just going to have to look at the numbers and see where I got them from. Here are the ratios for angle B: 

sin B = 3/5 cos B = 4/5 tan B = 3/4 

Once again just look at the numbers and reread the definitions to see where I came up with that stuff. But now that I told you a way of thinking of the ratios like opposite over hypotenuse there is one more way which should be easier and will also be discussed more later on. Here is a picture…notice how I am only dealing with one angle:

 

The little symbol in the corner of the triangle is a Greek letter called "theta"…its usually used to represent an unknown angle. Now with that picture we can think of sine, cosine and tangent in a different way:

sin (theta) = x/r cos (theta)= y/r tan (theta)= y/x -- and x <> 0

We will be using that form most of the time. Now although I may have skipped some kind of fundamentally important step (I'm hoping I did not) I can only think of one place to go from here: the unit circle. Becoming familiar with the unit circle will probably take the most work but make sure you do because it is very important. First let me tell you about radians just in case you do not know. Radians are just another way of measuring angles very similar to degrees. You know that there are 90 degrees in one-quarter of a circle, 180 degrees in one-half of a circle, and 360 degrees in a whole circle right? Well if you are dealing with radians there are 2p radians in a whole circle instead of 360 degrees. The reason that there are 2p radians in a full circle really is not all that important and would only clutter this "tutorial" more…just know that it is and it will stay that way. Now if there are 2p radians in a whole circle there are also p radians in a half, and p/2 radians in a quarter. Now its time to think about splitting the circle into more subdivisions than just a half or quarter. Here is a picture to help you out:

 

If at all possible memorize those values. You can always have a picture to look at like this one but it will do you well when you get into the more advanced things later on if you have it memorized. However that is not the only thing you need to memorize. Now you need to know (from memory if you have the will power) the sine and cosine values for every angle measure on that chart. 

OK I think I cut myself short on explaining what the unit circle is when I moved on to explaining radians. For now the only thing we need to know is that it is a circle with a radius of one centered at (0,0). Now the really cool thing about the unit circle is what we are about to discuss. I'm going to just pick some random angle up there on the graph…let's say…45 degrees. Do you see that line going from the center of the circle (on the chart above) to the edge of the circle? That point at which the line intersects the edge of the circle is very important. The "x" coordinate of that point on the edge is the cosine of the angle and the "y" coordinate is the sine of the angle. Very interesting huh? So lets find the sine and cosine of 45 degrees ourselves without any calculator or lookup tables. 

Well if you remember anything that I said at the beginning of this tutorial then you now know why I even mentioned it. In a right

triangle if there is an angle with a measure of 45 degrees the third angle is also 45 degrees. And not only that but the two legs of the triangle have the same length. So if we think of that line coming from the center of the circle at a 45-degree angle as a right triangle we can find the x- and y-position of where the line intersects…look at this picture:

If we apply some of the rules we learned about 45-45-90 triangles earlier we can accurately say that:

sqrt (2) sin 45 = -------- 2

sqrt (2) cos 45 = ---------- 2

Another way to think of sine is it's the distance from the x-axis to the point on the edge of the circle…you can only think of it that way if you are dealing with a unit circle. You could also think of cosine the same way except it's the distance from the y-axis to the point on the border of the circle. If you still do not know where I came up with those numbers look at the beginning of this tutorial for an explanation of 45-45-90 triangles…and why you are there refresh yourself on 30-60-90 triangles because we need to know those next.

Now lets pick an angle from the unit circle chart like 30 degrees. I'm not going to draw another picture but you should know how to form a right triangle with a line coming from the center of the circle to one of its edges. Now remember the rules that governed

the lengths of the sides of a 30-60-90 triangle…if you do then you can once again accurately say that:

1 sin 30 = ---- 2

sqrt (3) cos 30 = --------- 2

I was just about to type out another explanation of why I did this but it's basically the same as what I did for sine just above. Also now that I am rereading this I am seeing some things that may cause confusion so I thought I would try to clear up a few things. If you look at this picture (it's the same as the one I used a the beginning of all this) I will explain with a little bit more detail on how I arrived at those values for sine and cosine of 45-degrees:

 

Our definition of sine states that the sine of an angle would be the opposite side of the triangle divided by the hypotenuse. Well we know our hypotenuse is one since this a unit circle so we can substitute a one in for "c" and get this:

/ 1*sqrt(2) \ | ------------ | \ 2 /sin 45 = ------------------- 1

Which even the most basic understand of Algebra will tell us that the above is the same as: 

sqrt (2) sin 45 = -------- 2

Now if you do not get that look at it really hard until it comes to you…I'm sure it will hit you sooner or later. And instead of my wasting more time making a complete unit circle with everything on it I found this great link to one: http://www.infomagic.net/~bright/research/untcrcl.gif . Depending on just how far you want to go into this field of math as well as others like Calculus you may want to try and memorize that entire thing. Whatever it takes just try your best. I always hear people talking about different patterns that they see which helps them to memorize the unit circle, and that is fine but I think it makes it much easier to remember if you know how to come up with those numbers…that's what this whole first part of this tutorial was mostly about.

Also while on the subject I might as well tell you about the reciprocal trigonometric functions. They are as follow: 

csc (theta) = r/y sec (theta) = r/x cot (theta) = x/y 

Those are pronounced secant, cosecant, and cotangent. Just think of them as the same as their matching trigonometric functions except flipped…like this: 

sin (theta) = y/r - - - > csc (theta) = r/y cos (theta) = x/r - - - > sec (theta) = r/x tan (theta) = y/x - - - > cot (theta) = x/y 

That makes it a little bit easier to understand doesn't it? 

Well believe it or not that is it for an introduction to trigonometry. From here we can start to go into much more complicate areas. There are many other fundamentals that I would have liked to go over but this has gotten long and boring enough as it is. I guess I am hoping that you will explore some of these concepts and ideas on your own…you will gain much more knowledge that way as opposed to my sloppy words. 

Before I go… Before I go I want to just give you a taste of what is to come…this may actually turn out to be just as long as the above so go ahead and make yourself comfortable. First I want to introduce to you trigonometric identities, which are trigonometric equations that are true for all values of the variables for which the expressions in the

equation are defined. Now that's probably a little hard to understand and monotonous but I'll explain. Here is a list of what are know as the "fundamental identities":

Reciprocal Identities

1 csc (theta) = ---------- , sin (theta) <> 0 sin (theta)

1 sec (theta) = ---------- , COs (theta) <> 0 cos (theta)

1 cot (theta) = ---------- , tan (theta) <> 0 tan (theta)

Ratio Identities

sin (theta) tan (theta) = ------------ , cos (theta) <> 0 cos (theta)

cos (theta) cot (theta) = ------------- , sin (theta) <> 0 sin (theta)

Pythagorean Identities

sin^2(theta) + cos^2(theta) = 1

1 + cot^2(theta) = csc^2(theta)

1 + tan^2(theta) = sec^2(theta)

Odd-even Identities

sin (-theta) = -sin (theta)

cos (-theta) = cos (theta)

tan (-theta) = -tan (theta)

csc (-theta) = csc (theta)

sec (-theta) = sec (theta)

cot (-theta) = -cot (theta)

Now proving them…well that's gonna take a lot of room but here it goes. I'm only going to prove a few out of each category of identities so maybe you can figure out the others. Lets start with the reciprocal. Well if the reciprocal of a number is simply one divided by that number then we can look at cosecant (which is the reciprocal of sine) as:

1csc (theta) = ----- ----------------- >>> | If you multiply the numerator and the denominator by "r" you get: / y \ | |---- | < -- I hope you know | csc (theta) = r/y < -- Just like we said before. We just proved \ r / that is sine (theta) | an identity...I'll let you do the rest of them...

Now the ratio identities. If you think of tangent as y/x , sine as y/r , and cosine as x/r then check this out:

sin (theta) --- > y/r ytan (theta) = -------------- --- > ----- --- > Multiply top and bottom by "r" and you're left with --- > --- cos (theta) --- > x/r x

I'm going to save the proof for the Pythagorean Identities for another time. These fundamental identities will help us prove much more complex identities later on. Knowing trigonometric identities will help us understand some of the more abstract things…at least they are abstract to me. Once I am finished with this I am going to write another tutorial that will go into the somewhat more complex areas that I know of and these fundamental things I have just talked about are required reading.

I was going to go over some laws that can be very useful but my study plan tells me that I may not have provided enough information for you to understand it…therefore that will be something coming in the next thing I write.

Closing thoughts Well this concludes all the things that you will need to know before you start to do more complicated things. I was a bit brief with some things so if you have any questions or if you want me to go back and further explain something I implore you to e-mail me and I will do my best to clear up any confusion. Also I want to reiterate that

this is a very basic introduction to trigonometry. I hope you were not expecting to read this and learn all there is to know. Actually I have not really even mentioned Flash or the possibilities yet…and quite honestly there is not really anything to work with yet. However once I do start to mention Flash and the math that it will take to create some of these effects everyone sees it will almost be just like a review. When you sit down and want to write out a script it will be like merely translating everything you learned about trigonometry from a piece of paper into actionscript. 

If you want a little synopsis of what I plan on talking about in the next few things I write here you go:

- Trigonometry curves - More advanced look into trigonometry - Programmatic movement using trigonometry - Orchestrating it all into perfect harmony (pardon the cliché) 

Well that's it for me…until next time.

S U N D A Y , J U N E 1 8 , 2 0 0 6

The Definition of Cosine and SineIn this post I'd just like to briefly give the definition of cosine and sine, and also show a

simple property involving both of them.

Consider a right triangle containing an angle θ. All such triangles are just scalar multiples

of each other. Therefore the ratio of the adjacent side to the hypotenuse is a fixed value.

We'll call this ratio the cosine of θ. Similarly, we can define the sine ofθ as the ratio of the

opposite side to the hypotenuse

Now, we can show that

First we'll substitute

And by the Pythagorean Theorem

And so

PO S TE D B Y PH I L   AT   1 1 : 40 PM

2 COMMENTS:

Anonymous said...

Where's The Definition of Cosine and Sine?

MON MAR 19, 09:21:00 AM PDT

Phil said...

It's there .. it says that cosine of theta is defined as the ratio a / c, and sine is defined

as the ratio of b / c

I guess it could've been a bit more explicit .. possibly put those two statements in

their own stand-alone equations.

MON MAR 19, 09:44:00 AM PDT

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Trigonometry Definition Math Sheet

http://www.ecalc.com/pics/math/trig_definitions.gif

Trig Definition Math HelpRight Triangle DefinitionSine DefinitionCosine DefinitionTangent DefinitionCosecant DefinitionSecant DefinitionCotangent DefinitionUnit Circle DefinitionProperties of Trig FunctionsDomainRangePeriodInverse Trig FunctionsDefinition of Inverse Trig FunctionsDomain of Inverse Trig FunctionsRange of Inverse Trig Functions

Trig Definition Math HelpRight Triangle DefinitionTo define the trigonometric functions of an angle theta assign one of the angles in a right triangle that value. The functions sine, cosine, and tangent can all be defined by using properties of a right triangle. A right triangle has one angle that is 90 degrees. The longest side of the triangle is the hypotenuse. The side opposite theta will be referred to as opposite. The other side next to theta will be referred to as adjacent. The following properties exist:Sine Definition

Cosine Definition

Tangent Definition

Cosecant Definition

Secant Definition

Cotangent Definition

Unit Circle Definition

Properties of Trig FunctionsDomainThe possible angle input for each function is defined below:

RangeThe ranges of values possible for each of these functions are:

PeriodThe periods for each of these trig functions are:

Inverse Trig FunctionsDefinition of Inverse Trig FunctionsThe definitions of the inverse trig functions are:

Inverse Trig functions are also notated as:

Domain of Inverse Trig Functions

Range of Inverse Trig Functions