Sin, Cos & Tan (Edexcel IGCSE Further Maths)

Defining Sin, Cos and Tan

Spec point — 10B

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

  • $420^{\circ}$* *or $-750^{\circ}$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$\pm360^{\circ}$ you can keep going

    • So $420^{\circ}$ is 'all the way around' anticlockwise ($360^{\circ}$) and then another $60^{\circ}$ anticlockwise
    • Or $-750^{\circ}$ is 'all the way around twice' clockwise ($-720^{\circ}$) and then another $30^{\circ}$ clockwise

• 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

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.

Using The Unit Circle

Spec point — 10B

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 $\frac{3π}{2}$ rad)

  • Tangent (tan θ ) is positive in this quadrant

• The fourth quadrant is for angles between 270° and 360° ($\frac{3π}{2}$ rad and $2π$ 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 $\mathrm{sin}x=\frac{1}{2}$

• 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 $x = \sin ^{- 1} ( 1 \frac{1}{2} )$

    • 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 $\mathrm{cos}^{-1}$ to get the primary angle

    • Or in the first or fourth quadrants if using $\mathrm{sin}^{-1}$ or $\mathrm{tan}^{-1}$

  • If you are working with $\mathrm{sin}x=k$

    • 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 $\mathrm{cos}x=k$

    • 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 $\mathrm{tan}x=k$

    • 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 $\mathrm{cos}x=k$

    • This will be the quadrant which is vertically below the original quadrant

  • ... y coordinate if *solving $\mathrm{sin}x=k$

    • This will be the quadrant which is horizontally to the left of the original quadrant

  • ... gradient if solving $\mathrm{tan}x=k$

    • 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

Exam Hint

• 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 $\mathrm{cos}θ=0.8$ is θ = 0.6435 radians, correct to 4 decimal places.

Use this to find all other solutions in the interval $-2π\leqθ\leq2π$, giving your answers correct to 3 significant figures.

Trigonometry Exact Values

Spec point — 10B

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° ($2π$ radians) the trig values are all the same as for 0

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

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° ($\frac{π}{6}$ and $\frac{π}{3}$ radians) can be derived by using an equilateral triangle

  • See the Worked Example below

• Values for 45° ($\frac{π}{4}$ radians) can be derived using a right-angled isosceles triangle

  • See the following diagram

Exam Hint

• 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 $0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, $90^{\circ}$

  • The sine values run $0 , \frac{ \sqrt{1} }{2} ( = 1 \frac{1}{2} ) , \frac{ \sqrt{2} }{2} , \frac{ \sqrt{3} }{2} , \frac{ \sqrt{4} }{2} ( = 1 )$
  • The cosine values run $\frac{ \sqrt{4} }{2} ( = 1 ) , \frac{ \sqrt{3} }{2} , \frac{ \sqrt{2} }{2} , \frac{ \sqrt{1} }{2} ( = 1 \frac{1}{2} ) , 0$
  • Values for tangent can be found using $\mathrm{tan}θ=\frac{\mathrm{sin}θ}{\mathrm{cos}θ}$

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 $\frac{π}{3}$.