Sin, Cos & Tan (Edexcel IGCSE Further Maths)

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Amber

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Defining Sin, Cos and Tan

What is the unit circle?

The unit circle is a circle with radius 1 and centre (0, 0)
It helps us to define values for sin θ, cos θ and tan θ for all values of θ
- $420 °$ or $- 75 °$ doesn't make sense as an angle in a triangle or other shape
- But sin θ, cos θ and tan θ are defined for 'angles' like that
On the unit circle
- Angles are always measured from the positive x-axis and turn:
  - anticlockwise for positive angles
  - clockwise for negative angles
- After $\pm 360 °$ you can keep going
  - So $420 °$ is 'all the way around' clockwise ( $360 °$ ) and then another $60 °$ clockwise
  - Or $- 750 °$ is 'all the way around twice' anticlockwise ( $- 720 °$ ) and then another $30 °$ anticlockwise
It can be used to calculate trig values as coordinates (x, y) on the circle
- Make a right triangle with the radius as the hypotenuse
  - θ is the angle measured anticlockwise from the positive x-axis
  - (or clockwise for negative θ)
- The x-axis will always be adjacent to the angle, θ
SOHCAHTOA can then be used to find the values of sinθ, cosθ and tanθ
- As the radius is 1 unit
  - the x coordinate gives the value of cos θ
  - the y coordinate gives the value of sin θ
- Dividing the y coordinate by the x coordinate gives the value of tan θ
  - This is also the gradient of the line through the origin and the point on the unit circle
Unlike SOHCAHTOA this allows us to calculate sin, cos and tan for
- angles greater than 90° ( $\frac{π}{2}$ radians)
- negative angles

Unit circle

Worked example

The coordinates of a point on a unit circle, correct to 3 significant figures, are (0.629, 0.777). Find the angle with the positive x-axis, θ°, to the nearest degree.

efewCfDn_aa-sl-3-4-1-defining-sin-and-cos-we-solution-1

Using The Unit Circle

What are the properties of the unit circle?

The unit circle can be split into four quadrants
The first quadrant is for angles between 0 and 90°
- All three of sin θ, cos θ and tan θ are positive in this quadrant
The second quadrant is for angles between 90° and 180° ( $\frac{π}{2}$ rad and $π$ rad)
- Sine (sin θ ) is positive in this quadrant
The third quadrant is for angles between 180° and 270° ( $π$ rad and $\frac{3 π}{2}$ rad)
- Tangent (tan θ ) is positive in this quadrant
The fourth quadrant is for angles between 270° and 360° ( $\frac{3 π}{2}$ rad and $2 π$ rad)
- Cosine (cos θ ) is positive in this quadrant
Starting from the fourth quadrant (on the bottom right) and working anti-clockwise
- the positive trig functions spell out CAST
  - This is why it is often thought of as the CAST diagram
- You may have your own way of remembering this
  - A popular one starting from the first quadrant is All Students Take Calculus
To help yourself picture this try sketching all three trig graphs on one set of axes
- Look at which graphs are positive in each 90° section

How is the unit circle used to find additional solutions?

Trigonometric functions have more than one 'input' for each 'output'
- For example sin 30° = sin 150° = 0.5
This means that trigonometric equations have more than one solution
- Both 30° and 150° satisfy the equation $\sin x = \frac{1}{2}$
The unit circle can be used to find all solutions to trigonometric equations in a given interval
- Your calculator will only give one solution to an equation like
  - This solution is called the primary value
- However due to the periodic nature of the trig functions there there are an infinite number of solutions
  - You need to be able to find such additional values
- On the exam you will be given an interval in which the solutions should be found
  - This could either be in degrees or in radians
  - If you see π or some multiple of π then you must work in radians
The following steps can help you use the unit circle to find additional values
STEP 1
Draw the primary value angle using the x or y coordinates to help you
- This will be in the first or second quadrants if using to get the primary angle
  - Or in the first or fourth quadrants if using $\sin^{- 1}$ or $\tan^{- 1}$
- If you are working with
  - draw the line from the origin to the circumference of the circle
  - the point on the circumference is where the y coordinate is k
- If you are working with
  - draw the line from the origin to the circumference of the circle
  - the point on the circumference where the x coordinate is k
- If you are working with
  - draw the line from the origin to the circumference of the circle
  - such that the gradient of the line is k
- This will give you the angle which should be measured from the positive x-axis…
  - … anticlockwise for a positive angle
  - … clockwise for a negative angle
STEP 2
Draw the radius in the other quadrant which has the same...
- ... x coordinate if solving
  - This will be the quadrant which is vertically below the original quadrant
- ... y coordinate if solving
  - This will be the quadrant which is horizontally to the left of the original quadrant
- ... gradient if solving
  - This will be the quadrant which is diagonally opposite to the original quadrant
STEP 3
Work out the size of the second angle, measuring from the positive x-axis
- … anticlockwise for a positive angle
  - or clockwise for a negative angle
- Look at the given interval of solution values
  - This will help you to decide whether you need a negative or positive angle measure
STEP 4
Add or subtract (multiples of) either 360° or 2π radians to or from both values
- until you have all solutions in the required interval

sin, cos, and tan on the unit circle

Exam Tip

Practice sketching out the unit circle
- and using CAST to remember what is positive in what quadrant
This can help you to find all solutions to a problem in an exam question

Worked example

It is given that one solution of $\cos θ = 0.8$ is θ = 0.6435 radians, correct to 4 decimal places.

Use this to find all other solutions in the interval $- 2 π \leq θ \leq 2 π$ , giving your answers correct to 3 significant figures.

aa-sl-3-4-1-using-the-unit-circle-we-solution-2

Trigonometry Exact Values

What are exact values in trigonometry?

For certain angles the values of sin θ, cos θ and tan θ can be written exactly
- This means using fractions and surds
- You should be familiar with these values
  - and able to derive them using geometry
You should know the exact values of sin, cos and tan for angles of
- 0°, 30°, 45°, 60°, 90°, and 180° (in degrees)
- or $0, \frac{π}{6}, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}$ and $π$ (in radians)
These values are in the following table
- Note that for 360° ( $2 π$ radians) the trig values are all the same as for 0

Table of exact trigonometric values

How do I find the exact values of other angles?

The exact values for cos and sin can be seen on the unit circle
- They are the x and y coordinates respectively
- If using the unit circle coordinates to memorise the exact values
  - remember that cos (x value) comes before sin (y value)
The unit circle can also be used to find exact values of other angles using symmetry
If you know the exact values for an angle in the first quadrant
- you can draw the same angle from the x-axis in other quadrants to find values for other angles
Remember that the angles are measured anticlockwise from the positive x-axis
For example if you know that the exact value for sin 30° is 0.5
- draw a 30° angle from the horizontal axis in the three other quadrants
- measuring from the positive x-axis you have the angles of 150°, 210° and 330°
  - sine is positive in the second quadrant so sin 150° = 0.5
  - sine is negative in the third quadrant so sin 210° = - 0.5
  - sine is negative in the fourth quadrant so sin 330° = - 0.5
It is also possible to find the negative angles by measuring clockwise from the positive x-axis
- draw a 30° angle from the horizontal in the three other quadrants
- measuring clockwise from the positive x-axis you have the angles of -30°, -150°, -210° and -330°
  - sin is negative in the fourth quadrant so sin(-30°) = - 0.5
  - sin is negative in the third quadrant so sin(-150°) = - 0.5
  - sin is positive in the second quadrant so sin(-210°) = 0.5
  - sin is positive in the fourth quadrant so sin(-330°) = 0.5

exact values on the unit circle

How can I derive exact trig values?

Being able to derive basic exact trig values can help if you forget them
Values for 30° and 60° ( and radians) can be derived by using an equilateral triangle
- See the Worked Example below
Values for 45° ( radians) can be derived using a right-angled isosceles triangle
- See the following diagram

Exam Tip

It can be easy to muddle up exact trig values if you just try to remember them from a list,
Sketch the unit circle and/or trig graphs on your exam paper
- This can help you keep the trig functions and their values straight
Note that for $0 °$ , , , , $90 °$
- The sine values run $0, \frac{\sqrt{1}}{2} (= \frac{1}{2}), \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2} (= 1)$
- The cosine values run $\frac{\sqrt{4}}{2} (= 1), \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{1}}{2} (= \frac{1}{2}), 0$
- Values for tangent can be found using $\tan θ = \frac{\sin θ}{\cos θ}$

Worked example

Using an equilateral triangle of side length 2 units, derive the exact values for the sine, cosine and tangent of $\frac{π}{6}$ and $\frac{π}{3}$ .

Exact trig values from equilateral triangle

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Sin, Cos & Tan (Edexcel IGCSE Further Maths)

Revision Note

What is the unit circle?

What are the properties of the unit circle?

How is the unit circle used to find additional solutions?

What are exact values in trigonometry?

How do I find the exact values of other angles?

How can I derive exact trig values?

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Author: Amber