IB DP Maths: AA HL复习笔记3.4.2 Exact Values
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
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
Exam Tip
Worked Example
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