Exact Values (DP IB Analysis & Approaches (AA)): Revision Note

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 be able to derive the values using geometry

• You are expected to know the exact values of sin, cos and tan for angles of 0°, 30°, 45°, 60°, 90°, 180° and their multiples

  • In radians this is and their multiples

• The exact values you are expected to know are here:

How do I find the exact values of other angles?

• The exact values for sin and cos can be seen on the unit circle as the y and x coordinates respectively

  • If using the coordinates on the unit circle to memorise the exact values, remember that cos comes before sin

• The unit circle can also be used to find exact values of other angles using symmetry

• If you know the exact value for an angle in the first quadrant you can draw the same angle from the x-axis in any other quadrant to find other angles

• Remember that the angles are measured anticlockwise from the positive x-axis

• For example if you know that the exact value for is 0.5

  • draw the angle 30° from the horizontal in the three other quadrants

  • measuring from the positive x-axis you have the angles of 150°, 210° and 330°

    • sin is positive in the second quadrant so sin150° = 0.5

    • sin is negative in the third quadrant so sin210° = - 0.5

    • sin is negative in the fourth quadrant so sin330° = - 0.5

• It is also possible to find the negative angles by measuring clockwise from the positive x-axis

  • draw the angle 30° 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 are exact values in trigonometry derived?

• There are two special right-triangles that can be used to derive all of the exact values you need to know

• Consider a right-triangle with a hypotenuse of 2 units and a shorter side length of 1 unit

  • Using Pythagoras’ theorem the third side will be

  • The angles will be radians (90°), radians (60°) and radians (30°)

  • Using SOHCAHTOA gives…

    • Sin =               Sin =

    • Cos =                  Cos =

    • Tan =                Tan = =

• Consider an isosceles triangle with two equal side lengths (the opposite and adjacent) of 1 unit

  • Using Pythagoras’ theorem it will have a hypotenuse of

  • The two equal angles will be radians (45°)

  • Using SOHCAHTOA gives…

    • Sin

    • Cos

    • Tan = 1