Module: Construction Mathematics 2 Lecture 3

Simultaneous EquationsRefer to Mathematics Counts 4 book page 22. (attached copies).

Page 1 of 29

TrigonometryIntroduction to TrigonometryTrigonometry (from Greek trigonon "triangle" + metron "measure") Want to Learn Trigonometry? Here are the basics! Follow the links for more, or go to Trigonometry Index

Trigonometry ... is all about triangles.

A triangle has three sides and three angles

The three angles always add to 180°

Equilateral, Isosceles and ScaleneThere are three special names given to triangles that tell how many sides (or angles) are equal.

There can be 3, 2 or no equal sides/angles:

Equilateral TriangleThree equal sides Three equal angles, always 60°

Isosceles TriangleTwo equal sides Two equal angles

Scalene TriangleNo equal sides No equal angles

Page 2 of 29

Right Angled Triangle

A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side:

Adjacent is adjacent to the angle "θ", Opposite is opposite the angle, and the longest side is the Hypotenuse.

PerimeterThe perimeter is the distance around the edge of the triangle: just add up the three sides:

Area

The area is half of the base times height. "b" is the distance along the base "h" is the height (measured at right angles to the

base)

Area = ½ × b × h

The formula works for all triangles.Note: another way of writing the formula is bh/2

Example: What is the area of this triangle?

(Note: 12 is the height, not the length of the left-hand side) Height = h = 12Base = b = 20

Page 3 of 29

Area = ½ × b × h = ½ × 20 × 12 = 120The base can be any side, Just be sure the "height" is measured at right angles to the "base":

(Note: You can also calculate the area from the lengths of all three sides using Heron's Formula.)

Why is the Area "Half of bh"?Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (it would be a "parallelogram" actually), THEN the whole area would be bh (that would be for both triangles, so just one is ½ × bh), like this:

By slicing the new triangle and moving the sliced part to the other sideyou get a simple rectangle, whose area is bh.

Area of Triangles Without Right Angles

If You Know Base and HeightIt is easy to find the area of a right-angled triangle, or any triangle where we are given the base and the height.It is simply half of b times hArea = ½bh(The Triangles page tells you more about this).

Example: What is the area of this triangle?

Page 4 of 29

Height = h = 12Base = b = 20Area = ½ bh = ½ × 20 × 12 = 120If You Know Three Sides

There's also a formula to find the area of any triangle if we know the lengths of all three of its sides.This can be found on the Heron's Formula page.

If You Know Two Sides and the Included Angle

If we know two sides and the included angle (SAS), there is another formula (in fact three equivalent formulas) we can use.

Depending on which sides and angles we know, the formula can be written in three ways:Either Area = ½ab sin COr Area = ½bc sin AOr Area = ½ac sin BThey are really the same formula, just with the sides and angle changed.Example: Find the area of this triangle:

First of all we must decide what we know.We know angle C = 25º, and sides a = 7 and b = 10.

Page 5 of 29

So let's get going:

Start with: Area = ½ab sin C

Put in the values we know: Area = ½ × 7 × 10 × sin(25º)

Do some calculator work: Area = 35 × 0.4226...

Area = 14.8 to one decimal place

How to RememberJust think "abc": Area = ½ a b sin CHow Does it Work?Well, we know that we can find an area if we know a base and height:Area = ½ × base × height

In this triangle: the base is: c the height is: b × sin A

Putting that together gets us:Area = ½ × (c) × (b × sin A)Which is (more simply):Area = ½bc sin ABy changing the labels on the triangle we can also get:

Area = ½ab sin C Area = ½ca sin B

One more example:Example: Find How Much Land

Farmer Jones owns a triangular piece of land.The length of the fence AB is 150 m. The length of the fence BC is 231 m.The angle between fence AB and fence BC is 123º.How much land does Farmer Jones own?

First of all we must decide which lengths and angles we know:

Page 6 of 29

AB = c = 150 m, BC = a = 231 m, and angle B = 123º

So we use:Area = ½ca sinB

Start with: Area = ½ca sinB

Put in the values we know: Area = ½ × 150 × 231 × sin(123º) m2

Do some calculator work: Area = 17,325 × 0.838... m2

Area = 14,530 m2

Farmer Jones has 14,530 m2 of land

Heron's Formula

Area of a Triangle from SidesYou can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been know for nearly 2000 years.It is called "Heron's Formula" after Hero of Alexandria (see below)

Just use this two step process:

Step 1:Calculate "s" (half of the triangles perimeter) using:

Step 2: Then calculate the Area using:

Example: What is the area of a triangle where every side is 5 long?Step 1: s = (5+5+5)/2 = 7.5Step 2: A = √(7.5 × 2.5 × 2.5 × 2.5) = √(117.1875) = 10.825...

Pythagoras TheoremFor the next trigonometric identities we start with Pythagoras' Theorem:

Page 7 of 29

The Pythagorean Theorem states that, in a right triangle,the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):a2 + b2 = c2

Congruent TrianglesTriangles are congruent when they have exactly the same three sides and exactly the same three angles.What is "Congruent" ... ?It means that one shape can become another using Turns, Flips and/or Slides:

Rotation Turn!

Reflection Flip!

Translation Slide!

Congruent TrianglesIf two triangles are congruent they will have exactly the same three sides and exactly the same three angles.The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.Same SidesIf the sides are the same then the triangles are congruent.For example:

Page 8 of 29

is congruent to: and

because they all have exactly the same sides.But:

is NOT congruent to:

because the two triangles do not have exactly the same sides. Same AnglesDoes this also work with angles? Not always!Two triangles with the same angles might be congruent:

is congruent to:

only because they are the same sizeBut they might NOT be congruent because of different sizes:

is NOT congruent to:

because, even though all angles match, one is larger than the other.So just having the same angles is no guarantee they are congruent. Other CombinationsThere are other combinations of sides and angles that can work ...

Marking

Page 9 of 29

If two triangles are congruent, we often mark corresponding sides and angles like this:

is congruent to:

The sides marked with one line are equal in length. Similarly for the sides marked with two lines and three lines.The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs and three arcs.

Similar TrianglesTwo triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).These triangles are all similar:

(Equal angles have been marked with the same number of arcs)Some of them have different sizes and some of them have been turned or flipped.Similar triangles have:

all their angles equal corresponding sides have the same ratio

Corresponding SidesIn similar triangles, the sides facing the equal angles are always in the same ratio.For example:

Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.

Page 10 of 29

What are the corresponding lengths? The lengths 7 and a are corresponding (they face the angle marked with one arc) The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs) The lengths 6 and b are corresponding (they face the angle marked with three arcs)

Calculating the Lengths of Corresponding SidesIt may be possible to calculate lengths we don't know yet. We need to:

Step 1: Find the ratio of corresponding sides in pairs of similar triangles. Step 2: Use that ratio to find the unknown lengths.

Example: Find lengths a and b of Triangle S above.Step 1: Find the ratioWe know all the sides in Triangle R, and We know the side 6.4 in Triangle SThe 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R.So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is:6.4 to 8Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R.Step 2: Use the ratioa faces the angle with one arc as does the side of length 7 in triangle R.a = (6.4/8) × 7 = 5.6 b faces the angle with three arcs as does the side of length 6 in triangle R.b = (6.4/8) × 6 = 4.8 Done!

AnglesAngles (such as the angle "θ" above) can be in Degrees or Radians. Here are some examples:

Angle Degrees Radians

Right Angle 90° π/2

__ Straight Angle 180° π

Full Rotation 360° 2π

"Sine, Cosine and Tangent"

Page 11 of 29

The three most common functions in trigonometry are Sine, Cosine and Tangent. You will use them a lot!They are simply one side of a triangle divided by another.For any angle "θ":

Sine Function: sin(θ) = Opposite / Hypotenuse

Cosine Function: cos(θ) = Adjacent / Hypotenuse

Tangent Function: tan(θ) = Opposite / Adjacent

How to remember? Think "Sohcahtoa"! It works like this:

Soh... Sine = Opposite / Hypotenuse

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

...toa Tangent = Opposite / Adjacent

Other Functions (Cotangent, Secant, Cosecant)Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Cosecant Function: csc(θ) = Hypotenuse / Opposite

Secant Function: sec(θ) = Hypotenuse / Adjacent

Cotangent Function: cot(θ) = Adjacent / Opposite

Trigonometric and Triangle Identities

The Trigonometric Identities are equations that are true for all right-angled triangles.

The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle).

Example: What is the sine of 35°?

Page 12 of 29

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

Sine, Cosine and Tangent are often abbreivated to sin, cos and tan.

Example: 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):

Page 13 of 29

Sine sin(45°) = 1 / 1.414 = 0.707...

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

Tangent tan(45°) = 1 / 1 = 1

Unit Circle

The "Unit Circle" is a circle with a radius of 1.Being so simple, it is a great way to learn and talk about lengths and angles.The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.

Sine, Cosine and TangentBecause the radius is 1, you can directly measure sine, cosine and tangent.What happens when the angle, θ, is 0°?

cos 0° = 1, sin 0° = 0 and tan 0° = 0What happens when θ is 90°?

cos 90° = 0, sin 90° = 1 and tan 90° is undefined

Page 14 of 29

PythagorasPythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:x2 + y2 = 12

But 12 is just 1, so:x2 + y2 = 1 (the equation of the unit circle)Also, since x=cos and y=sin, we get:(cos(θ))2 + (sin(θ))2 = 1 (a useful "identity")

Important Angles: 30°, 45° and 60°You should try to remember sin, cos and tan for the angles 30°, 45° and 60°.Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, etc.These are the values you should remember!

Angle Sin Cos Tan=Sin/Cos

30° 1/√3 = √3/3

45° 1

60° √3

How To Remember?

To help you remember, think "1,2,3" :

sin(30°) = √1 = 1 (because √1 = 1)

Page 15 of 29

2 2

sin(45°) = √2

2

sin(60°) = √3

2

And cos goes "3,2,1"

cos(30°) = √3

2

cos(45°) = √2

2

cos(60°) = √1

= 1

(because √1 = 1)2 2

Just 3 Numbers

In fact, knowing 3 numbers is enough: 1

, √2

and √3

2 2 2

Because they work for cos as well as sin:

What about tan?Well, tan = sin/cos so you can calculate it like this:

an(30°) =

sin(30°)

=

1/2

=

1

cos(30°) √3/2 √3

But writing 1/√3 may cost you marks (see Rational Denominators), so instead use √3/3

tan(45°) =

sin(45°)

=

√2/2

= 1

cos(45°) √2/2

Page 16 of 29

tan(60°) =

sin(60°)

=

√3/2

= √3

cos(60°) 1/2

The Whole Circle For the whole circle we need values in every quadrant (with the correct plus or minus sign as per Cartesian Coordinates): Note that cos is first and sin is second, so it goes (cos, sin)

Example: What is cos(330°) ?

Page 17 of 29

Make a sketch like this, and you will see it is the "long" value: √3

2

And this is the same Unit Circle in radians.

Example: What is sin(7π/6) ?

Think "7π/6 = π + π/6", then make a sketch.You can then see it is negative and is the "short" value: −½

Footnote: where do the values come from?We can use the equation x2 + y2 = 1 to find the lengths of x and y (which are equal to cos and sin when the radius is 1):

Page 18 of 29

45 DegreesFor 45 degrees, x and y are equal, so y=x:x2 + x2 = 12x2 = 1x2 = ½x = y = √(½)

60 DegreesTake an equilateral triangle (all sides are equal and all angles are 60°) and split it down the middle.The "x" side is now ½,And the "y" side will be:(½)2 + y2 = 1¼ + y2 = 1y2 = 1-¼ = ¾y = √(¾)

30 Degrees30° is just 60° with x and y swapped, so x = √(¾) and y = ½

√(½) is also this:

And √(¾) is also this:

And here is the result (same as before):

Angle Sin Cos Tan=Sin/Cos

30° 1/√3 = √3/3

45° 1

Page 19 of 29

60° √3

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation.

When you need to calculate the function for an angle larger than a full rotation of 2π (360°) just subtract as many full rotations as you need to bring it back below 2π (360°):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° - 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

Likewise if the angle is less than zero, just add full rotations.

Example: what is the sine of -3 radians?

-3 is less than 0 so let us add 2π radians

-3 + 2π = -3 + 6.283 = 3.283 radians

sin(-3) = sin(3.283) = -0.141 (to 3 decimal places)

Graphs of Sine, Cosine and Tangent

Here are some nice graphs to look at ...

Page 20 of 29

Plot of Sine

The Sine Function has this beautiful up-down curve (which repeats every 2π radians, or 360°).

It starts at 0, heads up to 1 by π/2 radians (90°) and then heads down to -1.

Plot of Cosine

Cosine is just like Sine, but it starts at 1 and heads down until π radians (180°) and then heads up again.

Plot of Sine and Cosine

In fact Sine and Cosine are like good friends: they follow each other, exactly "π/2" radians, or 90°, apart.

Page 21 of 29

Plot of the Tangent Function

The Tangent function has a completely different shape ... it goes between negative and positiveInfinity, crossing through 0 (every π radians, or 180°), as shown on this plot.

At π/2 radians, or 90° (and -π/2, 3π/2, etc) the function is officially undefined, because it could be positive Infinity or negative Infinity.

Solving TrianglesA big part of Trigonometry is Solving Triangles. By "solving" I mean finding missing sides and angles.By "solving" I mean finding missing sides and angles.

If you know any 3 of the sides or angles ...... you can find the other 3

(Except for 3 angles, because you need at leastone side to find how big the triangle is.)

Six Different TypesIf you need to solve a triangle right now, then choose one of the six options below:

Which Sides or Angles do you know already? (Click on the image, or link)

AAA AAS ASA SAS SSA SSS

Page 22 of 29

Three Angles

Two Angles anda Side notbetween

Two Angles anda Side between

Two Sides andan Angle between

Two Sides andan Angle notbetween

Three Sides

Solving Triangles by Reflection

A 5ft ladder leans against a wall as shown.What is the angle between the ladder and the wall?

This is surprisingly easy to solve by using Reflection:

Here is the triangle with its reflectionTogether they make an equilateral triangle (all sides equal).

The angles in an equilateral triangleare all 60°

So the angle between the ladder and the wall is half of 60º= 30º

Finding Length

Page 23 of 29

We can use the same idea to find an unknown length.

Alex has a laser that measures distance.By standing some distance from the tree Alex measures42m to the top of the tree at an angle of 30º.What is the height of the tree?

Here is the triangle and its reflection:

Once again the triangle and its reflection make an equilateral triangle.So, we know the height of the tree must be half of 42m= 21m

The Law of SinesThe Law of Sines (or Sine Rule) is very useful for solving triangles:

It works for any triangle:

a, b and c are sides.A, B and C are angles.(Side a faces angle A, side b faces angle B and side c faces angle C).

So if you divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle CSure ... ?Well, let's do the calculations for a triangle I prepared earlier:

a/sin A = 8 / sin (62.2°) = 8 / 0.885... = 9.04...b/sin B = 5 / sin (33.5°) = 5 / 0.552... = 9.06...c/sin C = 9 / sin (84.3°) = 9 / 0.995... = 9.05...

Page 24 of 29

The answers are almost the same! (They would be exactly the same if I used perfect accuracy).So now you can see that:a/sin A = b/sin B = c/sin CHow Do I Use It?Let us see an example:Example: Calculate side "c"

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

Put in the values we know: a/sin A = 7/sin(35°) = c/sin(105°)

Ignore a/sin A (not useful to us): 7/sin(35°) = c/sin(105°)

Now we use our algebra skills to rearrange and solve:

Swap sides: c/sin(105°) = 7/sin(35°)

Multiply both sides by sin(105°): c = ( 7 / sin(35°) ) × sin(105°)

Calculate: c = ( 7 / 0.574... ) × 0.966...

Calculate: c = 11.8 (to 1 decimal place)

Finding an Unknown AngleIn the previous example we found an unknown side ...... but we can also use the Law of Sines to find an unknown angle.In this case it is best to turn the fractions upside down (sin A/a instead of a/sin A, etc):

Page 25 of 29

Example: Calculate angle B

Start with: sin A / a = sin B / b = sin C / c

Put in the values we know: sin A / a = sin B / 4.7 = sin(63º) / 5.5

Ignore "sin A / a": sin B / 4.7 = sin(63º) / 5.5

Multiply both sides by 4.7: sin B = (sin63º/5.5) × 4.7

Calculate: sin B = 0.7614...

Inverse Sine: B = sin-1(0.7614...)

B = 49.6º

Sometimes There Are Two Answers !There is one very tricky thing you have to look out for:Two possible answers.

Let us say you know angle A, and sides a and b.You could swing side a to left or right and come up with two possible results (a small triangle and a much wider triangle)Both answers are right!

This only happens in the "Two Sides and an Angle not between " case, and even then not always, but you have to watch out for it.Just think "could I swing that side the other way to also make a correct answer?" Example: Calculate angle R

Page 26 of 29

The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that's not a problem. We just use P,Q and R instead of A, B and C in The Law of Sines.

Start with: sin R / r = sin Q / q

Put in the values we know: sin R / 41 = sin(39º)/28

Multiply both sides by 41: sin R = (sin39º/28) × 41

Calculate: sin R = 0.9215...

Inverse Sine: R = sin-1(0.9215...)

R = 67.1º

But wait! There's another angle that also has a sine equal to 0.9215...Your calculator won't tell you this but sin(112.9º) is also equal to 0.9215... (try it!)So ... how do you discover the vale 112.9º?Easy ... take 67.1º away from 180°, like this:180° - 67.1° = 112.9°So there are two possible answers for R: 67.1º and 112.9º:

Both are possible! Each one has the 39º angle, and sides of 41 and 28.

Page 27 of 29

So, always check to see whether the alternative answer makes sense. ... sometimes it will (like above) and there will be two solutions ... sometimes it won't (see below) and there is one solution

We looked at this triangle before.As you can see, you can try swinging the "5.5" line around, but no other solution makes sense.So this has only one solution.

The Law of CosinesThe Law of Cosines (also called the Cosine Rule) is very useful for solving triangles:

It works for any triangle:

a, b and c are sides.C is the angle opposite side c

Let's see how to use it in an example:Example: How long is side "c" ... ?

We know angle C = 37º, a = 8 and b = 11

The Law of Cosines says: c2 = a2 + b2 - 2ab cos(C)

Put in the values we know: c2 = 82 + 112 - 2 × 8 × 11 × cos(37º)

Do some calculations: c2 = 64 + 121 - 176 × 0.798…

Page 28 of 29

Which gives us: c2 = 44.44...

Take the square root: c = √44.44 = 6.67 (to 2 decimal places)

Answer: c = 6.67

Page 29 of 29