Sin, Cos & Tan (Edexcel IGCSE Further Pure Maths)

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

  • or 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  you can keep going

    • So  is 'all the way around' clockwise () and then another clockwise

    • Or  is 'all the way around twice' anticlockwise () and then another  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° (radians)

  • negative angles  

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° ( rad and  rad)

  • Sine (sin θ ) is positive in this quadrant

• The third quadrant is for angles between 180° and 270° ( rad and  rad)

  • Tangent (tan θ ) is positive in this quadrant

• The fourth quadrant is for angles between 270° and 360° ( rad and  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

• 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  or   

  • 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

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 and   (in radians)

• These values are in the following table

  • Note that for 360° ( 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° ( 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