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Cogito

Trigonometry · Chapter 6 · Lesson 2

Practice Worksheet

Half Angle Formulas

Run the double angle formula backwards.

12 problems · about 21 minutes · F-TF.C.9

NameDate

Figure — use these to answer the problems

Compare one cycle against two: halving the angle stretches the wave, doubling squeezes it.
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y = 1 cos(1x) + 0
a, amplitude1
b, cycles1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. cos A = 0.6. What is sin²(A/2)?

    Answer

  2. Which angle decides the sign of sin(A/2)?

    • The half angle, A/2.
    • The original angle, A.

  3. cos A = 0. What is sin²(A/2)?

    Answer

  4. cos A = 0.8. What is sin²(A/2)?

    Answer

  5. cos A = 0.8. What is cos²(A/2)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. sin²(A/2) = 0.1 and cos²(A/2) = 0.9. What do they total?

    Answer

  2. cos A = −1. What is sin²(A/2)?

    Answer

  3. A = 90°. What is cos²(A/2), given cos 90° = 0?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the derivation of the half angle formula in order.

    Write 1 to 4 in the boxes to put these in order.

    • Begin with cos 2θ = 1 − 2sin²θ.
    • Rearrange to sin²θ = (1 − cos 2θ) ÷ 2.
    • Write A in place of 2θ, so θ becomes A ÷ 2.
    • Take the square root and choose the sign from the quadrant.

  2. cos A = 0.5, and A/2 lies in quadrant I. What is sin(A/2)?

    Answer

  3. The Half

    cos A = 0.5. What is sin²(A/2)?

    Answer

  4. The Sign

    A = 300°, so A/2 = 150°. Is sin(A/2) positive or negative?

    • Positive, because 150° is in quadrant II.
    • Negative, because 300° is in quadrant IV.