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## Sine and Cosine Rules

Sections:
• 0, Chapter 1, Introduction,

### Introduction

This unit extends the theme of finding angles and lengths in a right angle triangle to finding them in any triangle. After completing this unit you should

• understand how to use the sine and cosine rules in non right angled triangles
• be able to find the area of any triangles, given two sides and the included angle
• be able to use Heron's formula to calculate the area of any triangle, given the lengths of its three sides.

You have five sections to work through.

1. Sine Rule
2. Cosine Rule
3. Sine and Cosine
4. Application: Area of Any Triangle
5. Heron's Formula
6. Angles larger than $$90^{\circ}$$

• 0, Chapter 2, Sine Rule,

### Sine Rule

In the triangle $$ABC$$ shown below, the side opposite angle $$A$$ has length $$a$$, the side opposite angle $$B$$ has length $$b$$ and the side opposite angle $$C$$ has length $$c.$$

#### Remember

The sine rule states $$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$$

The sine rule can be used in any triangle to calculate:

• a side when two angles and an opposite side are known $$\left(AAS\right)$$
• an angle when two sides and an opposite angle are known $$\left(SSA\right)$$

The cosine rule states:
$$a^{2} = b^{2} + c^{2} - 2bc \; cos \; A$$
$$b^{2} = a^{2} + c^{2} - 2ac \; cos \;B$$
$$c^{2} = a^{2} + b^{2} - 2ab \; cos \; C$$
Note that when, for example $$A = 90^{\circ}$$ , the formula becomes, as expected, Pythagoras’ Theorem: $$a^{2} = b^{2} + c^{2}$$

The cosine rule can be used in any triangle to calculate:

• a side when two sides and the angle in between them are known $$(SAS)$$
• an angle when three sides are known $$(SSS)$$
• 1, Chapter 2, Task 1, Worked Example 1,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Find the unknown angles and side length of the triangle shown.

• #### Worked Example

Find two solutions for the unknown angles and side of the triangle shown.

• #### Worked Example

Find the unknown side and angles of the triangle shown.

• 0, Chapter 3, Cosine Rule,

### Cosine Rule

In the triangle $$ABC$$ shown below, the side opposite angle $$A$$ has length $$a$$, the side opposite angle $$B$$ has length $$b$$ and the side opposite angle $$C$$ has length $$c$$.

#### Remember

The cosine rule states:
$$a^2=b^2+c^2-2bc\cos A$$
$$b^2=a^2+c^2-2ac\cos B$$
$$c^2=a^2+b^2-2ab\cos C$$

The cosine rule can be used in any triangle to calculate:

• a side when two sides and the angle in between them are known $$\left(SAS\right)$$
• an angle when three sides are known $$\left(SSS\right)$$
• 1, Chapter 3, Task 1, Worked Example 2,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Find the unknown side and angles of the triangle shown.

• #### Worked Example

Find the shaded angle in the triangle shown.

• 1, Chapter 3, Task 2, Worked Example 3,

### Worked Example

• #### Worked Example

Find the unknown side and angles of the triangle shown.

• 0, Chapter 4, Sin and Cos,

## sin$$\theta$$ andcos$$\theta$$

The graphs of $$\sin \theta$$ and $$\cos \theta$$ for any angle are demonstrated iin the diagrams below which show the sine and cosine curves.

The graphs are examples of periodic functions. The graph repeats every $$360^\circ$$ (This is called the period , ie the period is $$360^\circ$$). In each period the maximum and minimum value of the functions are $$1$$ and $$-1$$ respectively.

Below are the values of sine and cosine for angles which often appear in questions.

 $$\theta$$ $$\sin \theta$$ $$\cos \theta$$ $$0^\circ$$ $$0$$ $$1$$ $$30^\circ$$ $$\frac{1}{2}$$ $$\frac{\sqrt{3} }{2}$$ $$45^\circ$$ $$\frac{1}{\sqrt{2} }$$ $$\frac{1}{\sqrt{2} }$$ $$60^\circ$$ $$\frac{\sqrt{3} }{2}$$ $$\frac{1}{2}$$ $$90^\circ$$ $$1$$ $$0$$

• 1, Chapter 4.1, Bearing,

### Bearing

#### Remember

A bearing is an angle in degrees measured clockwise from North.

#### Hot Tip

Some questions using sine and cosine are about objects (such as cars, ships, or joggers) travelling in certain directions for periods. In these questions you use sine or cosine to determine the eventual position of the object. They usually contain the concept of bearing described above.

Here is an example of such a question.

• 1, Chapter 4, Task 1, Worked Example 4,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

An oil tanker leaves Town X, and travels on a bearing of $$050^{\circ}$$ to Town Z, $$50 \; km$$ away.
The tanker then travels to Town Y, $$70 \; km$$ away, on a bearing of $$120^{\circ}$$.
a)   Find the distance of Y from X, giving your answer to 3 sig. figs.
b)   b. Calculate the bearing of $$X$$ from $$Y$$, giving your answer to the nearest degree

• #### Worked Example

Find the shaded angle in the triangle shown.

• 1, Chapter 4, Task 2, Exercise 1,

### Exercise

Here are some questions to check your progress

Exercise 1

1. Find the unknown angle marked $$\theta$$.
2. Find the unknown side marked $$a$$

Exercise 2

Find the unknown angles and sides.

Exercise 3

To calculate the height of a church tower, a surveyor measures the angle
of elevation of the top of the tower from two points 50 m apart.

1. Calculate the distance BC.
2. Hence calculate the height CD.

• 0, Chapter 5, Application: Area of Any Triangle,

### Application: Area of Any Triangle

An important application of trigonometry is that of finding the area of a triangle with side lengths a and b and included angle $$\theta$$

#### Area of a triangle when the lengths of two sides and the included angle are known,

The area (A) is given by $$A=\frac{1}{2} ab\sin \theta$$

We can prove this result by constructing the perpendicular line from point $$B$$ to the line $$AC$$.

Area $$= \frac{1}{2} \; \times \; base \times \; prependicular height$$
$$= \frac{1}{2} \times b \times p$$
where: $$p= a \; sin \; \theta$$
So Area $$= \frac{1}{2} \times b \times a \; sin \; \theta$$
$$= \frac{1}{2} \; a \; b \; sin \; \theta$$

• 1, Chapter 5, Task 1, Worked Example 5,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

The diagram shows  a circle of radius $$64\;cm$$. The length of the chord $$AB$$ is $$100\;cm$$.

1. Find the angle $$\theta$$ , to $$2 d.p$$.
2. Find the area of triangle $$OAB$$.
• #### Worked Example

Given that $$\sin \theta=\frac{\sqrt{3} }{2}$$, $$0^\circ\leq \theta\leq 90^\circ$$
Find  the area of triangle $$CDE$$ where

$$CD = 30\;units$$ and $$DE = 20\;units$$.

• 1, Chapter 5, Task 2, Exercise 2,

### Exercise

Here are some questions to check your progress

Exercise 1

Find the area of the shaded region

Exercise 2

Find the area of the shaded region

Exercise 3

In the diagram  ST = 5 cm , TW = 9 cm and $$STW = 52^\circ$$

Calculate

1. the length of SW
2. the area of $$\triangle$$ STW

• 0, Chapter 6, Heron's Formula,

### Heron's Formula

You have already met the formula for the area of a triangle when the lengths
of two sides and the included angle are known,

#### Formula for the area of a triangle when the lengths of all three sides are known.

$$A=\frac{1}{2} ab\sin C$$

We will use this result to find the formula for the area of a triangle when the lengths
of all three sides are known.

The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book,

The formula, known as Heron's formula, is given by

$$Area= \sqrt{s\left(s- a\right)\left(s- b\right)\left(s- c\right) }$$

$$S=\frac{a+ b+ c}{2}$$

• 1, Chapter 6, Task 1, Worked Example 6,

### Worked Example

• #### Worked Example

For the triangle shown, find

1. the area of the triangle,
2. angle $$\theta$$.

• 1, Chapter 6, Task 2, Exercise 3,

### Exercise

Here are some questions to check your progress

Exercise 1

Calculate the area of the triangle

Exercise 2

1. the area of the triangle,
2. the angle shown by $$\theta$$

• 0, Chapter 7, Angles Larger than $$90^{\circ}$$,

### Angles Larger than $$90^{\circ}$$

Let us explore how we define the trig rules for angles greater than $$90^{\circ}$$

First consider the $$x-y$$ plane is divided into four quadrants by the $$x$$ and $$y-axes$$ as shown in the picture below
The angle $$\theta$$ that a line $$OP$$ makes with the positive $$x-axis$$ lies between $$0^{\circ}$$ and $$360^{\circ}$$.

Angles between $$0^{\circ}$$ and $$90^{\circ}$$  are in the first quadrant.

Angles between $$90^{\circ}$$  and $$180^{\circ}$$ are in the second quadrant.

Angles between $$180^{\circ}$$ and $$270^{\circ}$$ are in the third quadrant.

Angles between $$270^{\circ}$$ and $$360^{\circ}$$ are in the fourth quadrant.

Angles larger than $$360^{\circ}$$ can be reduced to lie between $$0^{\circ}$$ and $$360^{\circ}$$ by subtracting multiples of $$360^{\circ}$$

#### Remember

The trigonometric formulae, $$\cos \theta$$ and $$\sin \theta$$, are defined for all angles between $$0^{\circ}$$ and $$360^{\circ}$$  as the coordinates of a point, $$P$$, where $$OP$$ is a line of length $$1$$, making an angle $$\theta$$ with the positive $$x-axis$$.

This is illustrated in the diagram below:

Here are some of the important values of $$\sin \theta$$ , $$\cos \theta$$ and $$\tan \theta$$ which we have meet before; you should already be familiar with these.

• 1, Chapter 7.1, Sin and Cos 1,

### Graphs of Sin$$\theta$$ and Cos$$\theta$$

The graphs of $$\sin \theta$$ and $$\cos \theta$$ for any angle are shown below.

The graphs are examples of periodic functions.  Each basic pattern repeats itself every $$360^{\circ}$$.  We say that the period is $$360^{\circ}$$.

For any angle, note that  $$\sin \left(90^{\circ}- \theta \right) = \cos \theta$$

The graph of $$\tan \theta$$ has period $$180^{\circ}$$. It is an example of a discontinuous graph.

The trigonometric equations  $$sin \theta =a$$ ,  $$\cos \theta =b$$ and $$\tan \theta=c$$ can have many solutions.

The inverse trig keys on a calculator (that is $$sin^{-1}$$, $$\cos^{-1},\tan^{-1}$$ give the principal value solution.

For $$\sin \theta=a$$ and $$\tan \theta=c$$ , the principal value solution is in the range:

$$-90^{\circ} \leq \theta \leq 90^{\circ}$$

For $$\cos \theta = b$$,  the principal value solution is in the range:

$$0 \leq \theta \leq 180^{\circ}$$

• 1, Chapter 7.2, Tan,

### Graphs of Tan$$\theta$$

The graph of tanθ has period $$180^{\circ}$$. It is an example of a discontinuous graph.

• 1, Chapter 7.3, Trigonometric Equations,

### Trigonometric Equations

The trigonometric equations $$sin \; \theta = a$$ , $$cos \; \theta = b$$ and $$tan \; \theta = c$$ can have many solutions.

The inverse trig keys on a calculator (that is $$sin^{-1}$$, $$cos^{-1}$$, $$tan^{-1}$$) give the principal value solution.

For $$sin \; \theta = a$$ and $$tan \; \theta = c$$ , the principal value solution is in the range:
$$-90^{\circ} \; \leq \; \theta \; \leq \; 90^{\circ}$$

For $$cos \; \theta = b$$, the principal value solution is in the range:
$$0^{\circ} \; \leq \; \theta \; \leq \; 180^{\circ}$$

• 1, Chapter 7.3, Task 1, Worked Example 7,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Find a)   $$cos \; 150^{\circ}$$
b)   $$sin \; 240^{\circ}$$

• #### Worked Example

If $$cos \; \theta = - \frac{1}{2}$$, how many values of the angle $$\theta$$ are possible for $$0 \leq \theta \leq 720^{\circ}$$. Find these values of $$\theta$$ .

• #### Worked Example

Use a calculator to solve the equation $$sin \theta = -0.2$$. Sketch the sin graph to show this solution. Give the principal value solution

scroll left to see another worked example

• 0, Chapter 8, Summary,

### Summary

the angle of elevation $$\alpha$$ is the angle made with the horizontal, as shown in the diagram
(angle $$\beta$$ is called the angle of depression). Angle $$\alpha = angle\; \beta$$ (alternate angles).

$$Angle\; \alpha=\;angle\;\beta$$ (alternate angles).

for any triangle ABC, the sine rule states
$$\frac{sin\;A}{a} = \frac{sin\;B}{b} = \frac{sin\;C}{c}$$
and the cosine rule states
$$c^{2}=a^{2}+b^{2} - 2 ab \cos C.$$
and similarly,
$$a^{2} = b^{2} + c^{2} - 2bc \cos A$$
$$b^{2} = c^{2} + a^{2} - 2ca \cos B$$

Interactive Exercises:
• Sine and Cosine Rules Interactive Exercises, https://www.cimt.org.uk/sif/trigonometry/t3/interactive.htm
• Sine and Cosine Rules , https://www.cimt.org.uk/sif/trigonometry/t3/interactive/s1.html
• Area of Any Triangle , https://www.cimt.org.uk/sif/trigonometry/t3/interactive/s2.html
• Heron's Formula, https://www.cimt.org.uk/sif/trigonometry/t3/interactive/s3.html
File Attachments:

## Pythagoras' Theorem

Sections:
• 0, Chapter 1, Introduction,

### Introduction

Many people have heard of Pythagoras' Theorem. In this unit we explain the theorem and give examples of its use.

After completing this unit you should;

• understand and be able to use Pythagoras' Theorem in right angled triangles
• be able to apply Pythagoras' Theorem when finding the length of an unknown side in a right angled triangle.

You have two sections to work through.

1. Pythagoras' Theorem
2. Further Work With Pythagoras’ Theorem
• 0, Chapter 2, Pythagoras Theorem,

### Pythagoras Theorem

Pythagoras' Theorem gives a relationship between the lengths of the sides of a right-angled triangle.

Pythagoras' Theorem states that:

In any right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides (the two sides that meet at the right angle).

$$Area A + Area B = Area C$$

#### Remember

In any right angle triangle  where $$c$$ is the length of the hypotenuse and $$b$$ and $$c$$ are the lengths of the triangle's other two sides

$$a^{2}+ b^{2}=c^{2}$$
• 1, Chapter 2.1, Finding the Hypotenuse,

### Finding the Hypotenuse

Hypotenuse

The hypotenuse is the side opposite the largest angle in a right-angled triangle.
In the diagram opposite "$$a$$" is opposite the 90 degree angle (largest angle).

See if you can spot the hypotenuse in the examplesbelow. The answers are at the bottom:

• 1, Chapter 2.1, Task 1, Worked Examples 1,

### Worked Examples

Use the slider to explore worked examples.

• #### Worked Example

Find the length of the hypotenuse of the triangle shown in the diagram.

• #### Worked Example

Find the length of the hypotenuse of the triangles shown in the diagrams below.

• #### Worked Example

Find the length of the side marked $$x$$ in this triangle.

• #### Worked Example

Find the length of the side marked $$x$$ in this triangle.

• #### Worked Example

Find the length of the side marked $$x$$ in this triangle.

• 1, Chapter 2.1, Task 2, Exercises 1,

### Exercises

Here are some questions to check your progress

Exercise

Find the length of the side x in each of the triangles below.

Exercise

Here is a problem solving question to solve.

• 0, Chapter 3, Further Work With Pythagoras’ Theorem,

### Further Work With Pythagoras’ Theorem

In this section we look at some slightly more complicated examples of applying Pythagoras's theorem.

Use the slider to explore worked examples.

• #### Worked Example

Find the length of the side marked $$x$$ in the diagram.

• #### Worked Example

Find the value of x as shown on the diagram.

• #### Worked Example

Verify Pythagoras' Theorem for the right-angled triangle opposite.

Solution

• 1, Chapter 3, Task 1, Exercises 2,

### Exercises

Here are some questions to check your progress.

Exercise

Find the value of $$x$$, giving your answer to 2 decimal places.

Exercise

Use Pythagoras Theorem to check that the following triangles contain a right angle.

• 0, Chapter 4, Summary,

### Summary

#### Pythagoras' Theorem

In any right angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides (the two sides that meet at the right angle).

Note that a right angle is indicated by a small 'box' at an angle which is $$90^{\circ}$$  as shown below

In a right angled triangle as labelled above you can use Pythagoras Theorem, namely

$$c^{2}=a^{2}+b^{2}$$      to find the length of $$c$$, the hypotenuse,

and  $$a^{2}=c^{2} - b^{2}$$  (or $$b^{2} = c^{2} - a^{2}$$)to find the length of $$a$$ (or $$b$$), given the lengths of $$b$$ and $$c$$ (or $$c$$ and $$a$$).

Interactive Exercises:
• Pythagoras' Theorem Interactive Exercises, https://www.cimt.org.uk/sif/trigonometry/t1/interactive.htm
• Pythagoras' Theorem, https://www.cimt.org.uk/sif/trigonometry/t1/interactive/s1.html
• Further Work with Pythagoras' Theorem, https://www.cimt.org.uk/sif/trigonometry/t1/interactive/s2.html
File Attachments:

## Trigonometric Ratios

Sections:
• 0, Chapter 1, Introduction,

### Introduction

This unit introduces the trigonometric ratios and shows how they can be used to find lengths and angles.

After studying this unit you should

• understand how to calculate the trigonometric functions, sine, cosine and tangent, in a right
angled triangle
• be able to find the length of a side in a right angled triangle, given one side and one angle
• be able to calculate sides and angles in trigonometric problems.

You have four sections to work through.

1. Sine, Cosine and Tangent
2. Finding Lengths in Right Angled Triangles
3. Finding Angles in Right Angled Triangles
4. Mixed Problems using Trigonometry

• 1, Chapter 1.1, Hypotenuse, Opposite and Adjacent,

#### Remember

A right angle triangle is a triangle witth a $$90$$ degree angle.

When working in a right angled triangle, the longest side is known as the Hypotenuse.

The other two sides are known as the Opposite and the Adjacent.
The adjacent is the side next to a marked angle, $$\theta$$ and the opposite side is opposite this angle.

• 0, Chapter 2, Sine, Cosine and Tangent,

### Sine, Cosine and Tangent

#### Remember

For a right-angled triangle, the sine, cosine and tangent of the angle $$\theta$$ are defined as:

$$\sin \theta = \frac{Opposite}{Hypotenuse}$$     $$\cos \theta = \frac{Adjacent}{Hypotenuse}$$     $$\tan \theta = \frac{Opposite}{Adjacent}$$

#### Top Tip

Remember the formula by learning the phrase SOHCAHTOA  which stands for sine, opposite, hypotenuse ; cosine, adjacent, hypotenuse; tangent, opposite, adjacent.

Note: $$\sin \theta$$ will always have the same value regardless of the size of the triangle.  (The same is true for $$\cos \theta$$ and $$\tan \theta$$)

• 1, Chapter 2, Task 1, Worked Example 1,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

For each triangle, state which side is:

1. the hypotenuse
3. the opposite

• #### Worked Example

Write down the values of $$\sin \theta , \cos \theta$$ and  $$\tan \theta$$ for the triangle shown.
Then use a calculator to find the angle in each case.

• 0, Chapter 3, Finding Lengths in Right Angled Triangles,

### Finding Lengths in Right Angled Triangles

Use the slider to explore worked examples.

• #### Worked Example

Find the length of the side marked $$x$$ in the triangle.

• #### Worked Example

Find the length of the side marked $$x$$ in the triangle.

• #### Worked Example

Find the length of the side marked $$x$$ in the triangle

• #### Worked Example

The diagram above, not drawn to scale, represents one face of the roof of a house in the shape of a parallelogram $$EFGH$$.

Angle $$EFI = 40^{\circ}$$ and $$EF =8\;m$$. $$EI$$ represents a rafter placed perpendicular to $$FG$$ such that $$IG = 5\;m$$.

Calculate $$FI, EI$$ and the $$area$$ of the parallelogram giving your answers to $$3$$ significant figures.

• 0, Chapter 4, Finding angles in Right Angled Triangles,

### Finding angles in Right Angled Triangles

If the lengths of any two sides of a right-angled triangle are known, then sine, cosine and tangent can be used to find the angles of the triangle.

In these examples we introduce some notation namely we use $$sin^{-1}\left(x\right)\ to mean the angle whose sine is \(x$$. Similar notation is used for cosine and tangent.

• #### Worked Example

Find the angle marked $$\theta$$ in the triangle shown.

• #### Worked Example

Find the angle marked $$\theta$$ in the triangle shown.

• #### Worked Example

Find the angle marked $$x$$ in the triangle shown.

• 0, Chapter 5, Mixed problems using Trigonometry,

### Mixed problems using Trigonometry

Use the slider to explore worked examples.

• #### Worked Example

$$ABC$$ is a right angled triangle. $$AB$$ is of length $$4\;m$$ and $$BC$$ is of length $$13\;m$$.

1. Calculate the length of $$AC$$.
2. Calculate the size of angle $$ABC$$.

• #### Worked Example

The diagram shows a roofing frame $$ABCD$$.
$$AB =7\;m$$, $$BC=5\;m$$, and angle $$ABD =$$ angle $$DBC = 90^{\circ}$$

See if you can work out the details of this solution yourself.

1. Calculate the length of $$AD$$.
2. Calculate the size of angle $$DCB$$.
• 0, Chapter 6, Summary,

### Summary

In a right angled triangle, given an angle $$\theta$$ and the length of one side, you can use

$$\sin \theta = \frac{opp}{hyp} = \frac{a}{c}$$

$$\cos \theta = \frac{adj}{hyp} = \frac{b}{c}$$

or

$$\tan \theta = \frac{opp}{adj} = \frac{a}{b}$$

to find the lengths of the other two sides and to find angles, using the SHIFT button on a calculator
[Note that calculators should be in degree mode for trigonometric calculations.]
For example,
(i) given $$c$$ and angle $$\theta$$, $$a=c \sin \theta$$, $$b = \cos \theta$$
(ii) given $$a$$ and angle $$\theta$$, $$c=\frac{a}{sin \theta}$$, $$b=\frac{a}{tan \theta}$$, etc.

The angle of elevation $$\left(a\right)$$ is the angle made with the horizontal, as shown in the diagram (angle $$\beta$$ is called the angle of depression).
$$Angle\;α = angle \beta$$ (alternate angles).

Important values of sin, cos and tan are shown in the table below.

Interactive Exercises:
• Interactive Exercises - Trigonometric Ratios, https://www.cimt.org.uk/sif/trigonometry/t2/interactive.htm
• Sine, Cosine and Tangent, https://www.cimt.org.uk/sif/trigonometry/t2/interactive/s1.html
• Finding Lengths in Right Angled Triangles, https://www.cimt.org.uk/sif/trigonometry/t2/interactive/s2.html
• Finding Angles in Right Angled Triangles, https://www.cimt.org.uk/sif/trigonometry/t2/interactive/s3.html
• Mixed Problems Using Trigonometry, https://www.cimt.org.uk/sif/trigonometry/t2/interactive/s4.html

## Further Trigonometry

Sections:
• 0, Chapter 1, Introduction,

### Introduction

After studying this unit, you should;

• understand the concept of radian measure and be able to change angle measures to and from degrees and radians
• be able to use the formulae for arc length and sector area, working in radians
• be able to extend the sine, cosine and tangent functions for any values
• be able to solve trigonometrical equations
• understand the properties of trig functions.

The unit is divided into the following five sections.

2. Arc and Sector
3. Sine, Cosine and Tangent Functions
4. Solving Trigonometrical Equations
5. Properties of Trignometrical Functions

• 0, Chapter 2, Radian Measure,

Radians are an alternative unit of measure for angles, often used in trigonometry. They relate the arc length of a sector to its angle. Here is an explanation of how radians are worked out. One radian corresponds to the angle which gives the same arc length as the radius of the circle.

Remember for any circle with radius r

Circumference of the circle = 2𝜋𝑟

Here is a drawing of the circle showing a sector

The arc length $$PQ= \frac{\theta}{360} \times 2 \pi r$$
If $$PQ=r$$
Then $$r=\frac{\theta}{360} \times 2 \pi r$$
Rearranging, $$\theta = \frac{360}{2 \pi}$$
$$\theta = \frac{180}{\pi}$$
One radian $$= \frac{180}{\pi}$$

This is not a very convenient definition, so it is usual to note that:
$$\pi$$ radians $$=180$$ degrees
1 degree $$\equiv \frac{\pi}{180}$$

#### Remember

One radian $$= \frac{180}{\pi}$$

1 degree $$\equiv \frac{\pi}{180}$$

• 1, Chapter 2, Task 1, Worked Example 1,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Convert the following angles in degrees to radians.
a)   $$90^{\circ}$$
b)   $$720^{\circ}$$

• #### Worked Example

Convert the following angles in radians to degrees.
a)   $$\frac{2 \pi }{3}$$ radians
b)   $$\frac{\pi}{12}$$ radians

• 0, Chapter 3, Arc and Sector Area,

### Arc and Sector Area

Ever wanted to prove that your friend has cut a bigger slice of pizza for themselves than they did for you? Here is how to do it. In fact you can work out the area of pizza slice as well the length of your slice on the curve, known as the arc. Go on, amaze and shame your mate!

Let’s start, as always in trigonometry, with a diagram. In this case a pizza which a perfect circle (congratulations to the chef).

From our formula , Arc length $$PQ = \frac{\theta}{360} \times 2 \pi r = \frac{\pi r \theta}{180}$$
If $$\theta$$ is measured in radians
are length $$PQ = \frac{\theta}{2 \pi} \times 2 \pi r =\ r \; \theta$$

The region between the two radii and the arc is called a sector of the circle.
Area of a sector $$POQ = \frac{\theta}{360} \times \pi r^{2}$$
If $$\theta$$ is measured in radians
area of a sector $$POQ = \frac{\theta}{2 \pi} \times \pi r^{2} = \frac{1}{2} \theta r^{2}$$

In summary, when $$\theta$$ is measured in radians,
Arc length $$= r \; \theta$$ and Sector area $$= \frac{1}{2} \theta r^{2}$$

#### Remember

When $$\theta$$ is measured in radians the arc length  and sector area in a circle with radius $$r$$ is given by
Arc length $$= r \; \theta$$   and
Sector area $$= \frac{1}{2} \theta r^{2}$$

• 1, Chapter 3, Task 1, Worked Example 2,

### Worked Example

• #### Worked Example

a)   A silver pendant is made in the form of a sector of a circle as shown below. If the radius is $$3\;cm$$, what is the angle, $$\theta$$, in radians so that the area is $$6 cm^{2}$$ ?

b)   Another pendant has the same total perimeter, but with radius $$2.5\;cm$$. What is the required angle, $$\theta$$, in radians?

• 0, Chapter 4, Sine, Cosine and Tangent Functions,

### Sine, Cosine and Tangent Functions for Angles greater than 90

Previously we have used sine, cosine and tangent with angles between $$0^{\circ}$$ and $$90^{\circ}$$ We are now going to apply them to other angles.

Angles of more than $$90^{\circ}$$ can be defined as the angle $$\theta$$ made between a rotating 'arm' OP and the positive x axis, as shown below.

It is possible to define angles of more than $$360^{\circ}$$ in this way, or even negative angles, as shown in the diagram below.

#### Remember

If the length of $$OP$$ in the diagram above is $$1$$ unit, then the sine, cosine and tangent of any angle is defined in terms of the $$x$$ and $$y$$ co-ordinates of the point $$P$$ as follows;
$$sin \; \theta = y$$  $$cos \; \theta = x$$  $$tan \; \theta = \frac{y}{x}$$

• 1, Chapter 4.1, Sine, Cosine,

### Sine, Cosine

The functions $$y=sin \; x$$ and $$y=cos \; x$$ can be plotted on a graphic calculator or computer, or indeed by hand, between $$0^{\circ}$$ and $$360^{\circ}$$. THe functions are shown below.

The maximum and minimum values of the function $$y$$ are $$1$$ and $$-1$$.
Both functions, $$sin \; x$$ and $$cos \; x$$, lie between $$\pm 1$$.
The amplitude is the height from the $$x$$ -axis to the peak (or trough).
The amplitude for both sine and cosine functions is $$1$$.

For the range $$360^{\circ} \leq x \leq 720^{\circ}$$, the pattern will repeat itself, and similarly for negative values.

A period is the length of one cycle before the pattern repeats. The period for both sine and cosine functions is $$360^{\circ}$$.

The graphs are illustrated below.

You can see that the the function $$y=sin \; x$$ is an odd function as $$y(-x) = -y(x)$$ whilst $$y=cos \; x$$ is an even function as $$y(-x)=y(x)$$.

Note that:
$$cos \; \theta = sin \; (90 - \theta^{\circ})$$
$$sin \; \theta = cos \; (90 - \theta^{\circ})$$

• 1, Chapter 4.1, Task 1, Worked Example 3,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Show that $$sin \; 30^{\circ} = \frac{1}{2}$$

• #### Worked Example

From the graphs, find the following values shown on the graphs

a)   $$sin \; 150^{\circ}$$

b)   $$sin \; 210^{\circ}$$

c)   $$sin \; 390^{\circ}$$

d)   $$sin \; (-30^{\circ})^{\circ}$$

e)   $$cos \; 60^{\circ}$$

• 1, Chapter 4.2, Tangent Function,

### Tangent Function

The last function to consider here is $$tan \; x$$, which has rather different properties from $$sin \; x$$ and $$cos \; x$$.

We can use a graphic calculator or computer to sketch the function $$y=tan \; x$$ for $$x$$ in any range. The function is shown below.

The period of the function is the distance between any two consecutive asymptotes (dotted lines); the period for $$y=tan \; x$$ is $$180^{\circ}$$.

Unlike the sine and cosine functions, tan $$x$$ is not bounded by $$\pm 1$$.

In fact, as $$x$$ increases to $$90^{\circ}$$, $$tan \; x$$ increases without limit.

• 1, Chapter 4.3, Remember,

### Remember

Here is a useful reminder of some of the values of the trig functions

• 1, Chapter 4, Task 2, Worked Example 4,

### Worked Example

• #### Worked Example

Using the values:
$$tan \; 45^{\circ} = 1 \;$$ and $$tan \; 30^{\circ} = \frac{1}{\sqrt{3}}$$
find the following without a calculator:

a)   $$tan \; 135^{\circ}$$

b)   $$tan \; (-45^{\circ})$$

c)   $$tan \; 315^{\circ}$$

d)   $$tan \; (-30^{\circ})$$

e)   $$tan \; 150^{\circ}$$

f)   $$tan \; 60^{\circ}$$

• 0, Chapter 5, Solving Trigonometrical Equations,

### Solving Trigonometrical Equations

Trigonometric functions can be used to model physical phenomena but applying these functions to problems in the real world will often result in a trigonometric equation to solve.

In this section, you will consider some simple equations to solve. The worked examples in the next section will highlight the difficulties that can occur.

Here is a HOT TIP to remember for solving trigonometrical equations

#### Remember

Whenever solving a trigonometrical equation also draw a sketch of the functions. This will help you identify multiple solutions to the equations.

• 1, Chapter 5, Task 1, Worked Example 5,

### Worked Example

Use the slider to explore worked examples.

• #### Worked Example

Solve the equation below
$$3 \; cos \; x=-0.6$$ for $$0^{\circ} \leq x \leq 360^{\circ} \;$$ giving the answer to 1 d.p.

• #### Worked Example

Solve the equation tan $$x=-2$$ where $$x$$ is measured in radians and $$-2 \pi \leq x \leq 2 \pi$$. Give your answers to $$2$$ decimal places. (Set your calculator to radian mode.)

• 0, Chapter 6, Properties of Trigonometrical Functions,

### Properties of Trigonometrical Functions

We will explore some of the properties of trigonometrical functions by looking at some worked examples. You will see from the worked examples the importance of drawing a diagram

Use the slider to explore worked examples.

• #### Worked Example

Find all solutions between $$0^{\circ}$$ and $$360^{\circ}$$ of the equation $$sin \; x = 2 \; cos \; x$$ to 2 decimal points.

• #### Worked Example

Find all solutions between $$0^{\circ}$$ and $$360^{\circ}$$ of the equation $$cos \; x \; sin \; x = 3 \; cos \; x$$

• 0, Chapter 7, Summary,

### Summary

One radian $$= \frac{180}{\pi}$$

This is not a very convenient definition, so it is usual to note that:

$$\pi$$ radians $$=180$$ degrees

$$1 \; degree \equiv \frac{\pi}{180}$$

and note that

$$45^{\circ} = \frac {\pi}{4}$$

$$90^{\circ} = \frac {\pi}{2}$$

$$180^{\circ} = \pi$$,

$$270^{\circ} = \frac{3\pi}{2}$$

$$360^{\circ} = 2 \pi$$

Arc length

arc length, $$PQ = r \theta$$

Sector area

sector area $$OPQ = \frac {1}{2}r^{2} \theta$$

Sine, cosine and tangent functions for all angles

Properties of trig functions

Degrees

$$\cos \left(x\right) = \sin \left(90^{\circ} - x\right)$$

$$\sin \left(x\right) = \sin \left(90^{\circ} - x\right)$$

$$\cos \left(180^{\circ} - x\right) = - \cos \left(x\right)$$

$$\sin \left(180^{\circ} - x\right) = + \sin \left(x\right)$$

$$\tan x = \frac {\sin x}{\cos x}$$

$$\sin^{2} x + \cos^{2} x =1$$

$$\cos \left(x\right) = \sin \left(\frac{\pi}{2} - x\right)$$

$$\sin \left(x\right) = \cos \left(\frac{\pi}{2} - x\right)$$

$$\cos \left(\pi - x\right) = -\cos \left(x\right)$$

$$\sin \left(\pi - x\right) = \sin \left(x\right)$$

Interactive Exercises:
• Further Trigonometry Interactive Exercises, https://www.cimt.org.uk/sif/trigonometry/t4/interactive.htm