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MTH101 Introduction toMathematicsLecture 6 - Trigonometry

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Lecture Objectives Apply Pythagoras Theorem to right triangles to find one

unknown side when two sides are known. Use the trigonometric ratios for right triangles in calculations

to find unknown sides and angles. Know the trigonometric ratios Sin, Cos and Tan. Given an angle, correctly identify the names of each of the

sides of a right triangle. Use these trig ratios to find an unknown side or angle in a right

triangle. Interpret word questions and produce appropriate diagrams

which facilitate mathematical solution using trig ratios. Understand the periodic nature of the trig functions. Textbook Chapter 4 pages 108-129, Chapter 8 pages 231-

250, Chapter 10 pages 288-299

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Pythagoras Theorem Pythagoras theorem applies to right

triangles only.

The hypotenuse is the longest sideand is opposite the right angle.

Pythagoras Theorem in words states

that the length of the Hypotenusesquared is equal to the sum of thesquares of the other two sides. Interms of the letters representing thesides of the triangle, the theoremstates: a2 +b2 = c2.

a

bc (Hypotenuse)

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Pythagoras Theorem Examples Example: Calculate the length of the

hypotenuse for the triangle shown.

Solution: Always identify the side which is the

hypotenuse first. In this case it is markedas c.

State Pythagoras Theorem in terms of the

information contained within theproblem.

Ensure that if the lengths involved in the

problem are given with units that theappropriate units are shown for the

answer. Also note that all of the sidesmust be in the same unit before thetheorem is applied. If one side is in

6

5c (Hypotenuse)

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Pythagoras Example Example: Find the unknown side

for the right triangle shown.

Solution: Identify which side is the

hypotenuse. In this case it isthe side with length 9.

State Pythagoras Theorem usingthe sides as required with thehypotenuse on its own.

Solve for the value of c.

95

c

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Practical Example Example: A man sitting in a boat knows

that he is 600m from the bottom of acliff at the waters edge. If the directdistance from the man to the top of thecliff is 800m, how high is the cliff?

Solution: IMPORTANT: Draw a diagram to

illustrate the information you havein the question. In this case thediagram you draw would looksomething like this.

Identify which side is the hypotenuse

and substitute into PythagorasTheorem.

Always answer word questions in

words.

600m

800mx m

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Trig Functions with a Calculator Make sure your calculator is set-up correctly.

To set the mode on your calculator, followthese steps.

Set the calculator to Calculation and Mathmode as normal. Select the Shift Setup button combination. You will see the following screen. Notice that you have options which allow the

setting of degrees or radians . Select the number 3 for degrees

and 4 for radians. You will see the screen indicate the change. In TPP104, we will normally work in

degrees, but the methods we use areapplicable to both.

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Calculate the value of a trig function

Example 1 Example 2

Example:Calculate the value of

Solution: Enter the followingkeystrokes into thecalculator to calculate theanswer.

Note that the closing

bracket is not reallyrequired, but you need tobe careful with using

brackets. You will seethe answer .9563 appear

Example: If the Sin of an angle is .766, what is the angle? (This is

the same as saying: What is thevalue of )

Solution: Note that this is the reverse of

the previous example. Ingeneral, if you want to do

the reverse of any functionon the calculator, yousimply precede the normalbutton used with the Shiftbutton.

You will see the answer of

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Trigonometry Exercises Exercises:

Answers: 1. A. 34.36 B. 22.58 C. 5.33

2. A. 42o B. 58o C. 28o D. 51o

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Trigonometry with right triangles The trig functions may be used in

calculations relating to the sidesand angles of triangles.Depending on which sides andangles you have, you use different

functions.

Before we use the functions, youmust be able to identify the namesof the sides of a triangle, whengiven an angle.

Working from the angle in a righttriangle, we identify:

The Hypotenuse (which is oppositethe right angle)

The Opposite (which is the sideopposite the angle)

The adjacent (which is the sidebeside the angle which is NOTthe hypotenuse)

Hypotenuse

Opposite

Adjacent

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Trigonometry Ratios for a right triangle

There are three trig ratios we will use (there are others) Sine, Cosine and Tangent

For ease of use these are reduced to Sin, Cos and Tan. Note that Sin is pronounced as in Sign, not the Sin that istalked about in the bible.

The rules for the trig ratios are:

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Remembering the Ratios Use SOH CAH TOA

Each group of three letters is a rule. The firstletter is the trig ratio, the last two letters are thesides you use.

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Trigonometric Ratio Example Example: Calculate the length of

the unknown side in the righttriangle shown.

Solution: Identify the sides in relation to

the angle involved. In thiscase, x is the Opposite and 10is the Hypotenuse.

Identify the correct trig function

to use with these sides. In thiscase

Substitute in the equation andsolve for the unknown.

30

10x

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Trigonometric Ratio Selection Remember the first step in correctly solving

trigonometric ratio problems is to correctlyidentify the sides you have in the problem.

Identify the sides i.e. Hypotenuse, Opposite or

Adjacent in each of the following triangles:

a

bc

c

b

a

Hypotenuse Opposite Adjacent

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Trig with Triangles 1 Exercises: Find the

length of the

unknown side in theright triangle.

Solution: Identify the name of

the sides Identify the trig

ratio required Substitute and solve

25

1.5m

x

Hypotenuse

Adjacent

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Trig with Triangles 2 Exercises: Find the

length of the

unknown side in theright triangle.

Solution: Identify the name of

the sides Identify the trig

ratio required Substitute and solve

65

2.5mx

Hypotenuse

Opposite

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Trig with Triangles 3 Exercises: A ladder leans

against a wall. If the ladderis 5m long and ispositioned so that itreaches 3m up the wall,what is the angle the laddermakes with the ground?

Solution: Draw a diagram!!! Identify the sides. Identify the trig ratio. Substitute and solve. Give the answer in words. The angle between the

ladder and the group is

38.87o

.

3m

5m

x

Let x = angle ladder toground

HypotenuseOpposite

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Trig Example 2 Example: A ladder leans against the top of a

wall and makes an angle with the ground of35o. If the foot of the ladder is 2.5m fromthe wall, how long is the ladder?

Solution: Always draw a diagram to illustrate the

information you are given.

Identify the names of the sides in relation

to the angle you have. In this case,

2.5m is the adjacent and x is thehypotenuse.

Identify which trig function you need to

use with these two sides. In this case itis Cos. Then substitution in the Cosformula and calculate the answer asfollows.

35

2.5m

x

Let x = length of the ladder

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Angles of Elevation and Depression Angles of elevation and depression are both

measured from the horizontal.

Angle of elevation:

Angle of depression:

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Angle of Depression Example Example: A pilot in a plane

observes that the angle ofdepression to an airport is 40o.His instruments indicate that he

is flying at a height of 7,500m.What is the ground distance fromthe plane to the airport?

Solution: Draw a diagram to illustrate the

information provided.

We need an angle in the triangle. Identify the sides involved. Identify the trig ratio required. Substitute and solve. Answer in words: Ground

distance from the plane to theairport is 8,938m.

Angle ofDepression

50o

7,500m

40o

x

Let x = ground distanceplane to airport in m

Opposite

Adjacent

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Radians or Degrees

1

1

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The Periodic Nature of Sin and Cos The graphs of the trig functions Sin and Cos are

quite interesting. Consider the data grid shown

below which shows a series of values for anglesand their corresponding Sin and Cos values.

Graphing these values gives the following result

Angle 0 45 90 135 180 225 270 315 360Sin 0 .7071 1 .7071 0 -.7071 1 -.7071 0Cos 1 .7071 0 -.7071 1 -.7071 0 .7071 1

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The Periodic Nature of Sin and CosAngle 0 45 90 135 180 225 270 315 360Sin 0 .7071 1 .7071 0 -.7071 1 -.7071 0Cos 1 .7071 0 -.7071 1 -.7071 0 .7071 1

Angle

Sin/Cos

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The General Form 1 y = sin x is the simplest form;

The general form is y= a sin(bx+c), where a, b and

c are some constants. You need to understand how

the presence of each constantaffects the shape of thegraph.

Properties of y = sin x

Amplitude = 1;

Period = 2

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The General Form 2 a -> Amplitude

a > 1 increases amplitude

0 > a < 1 decreases amplitude

b -> Period

c -> Phase Shift

X 2

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Drawing trig functions Sketch Sketch

1/2

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Inverse Trig Functions You just need to know of the existence of these.

Remembertheseversions

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Lecture Overview Pythagoras Theorem;

Trig functions in right triangles;

Periodic nature of trig functions; General form: y = a sin(bx + c)

Sketching trig functions