6. Trigonometric Identities and More Equations

Use the given information to find each of the following.

sin A/2, given cos A/2 = - 3, 90° < A < 180°

Use the given information to find each of the following.

sin y, given cos 2y = -1/3 , π/2 < y < π

Use the given information to find each of the following.

sin 2x, given sin x = 0.6, π/2 < y < π

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x

Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Match each expression in Column I with its value in Column II.

8. tan (-π/8)

Match each expression in Column I with its value in Column II.

10. cos 67.5°

Use a half-angle identity to find each exact value.

sin 195°

Use a half-angle identity to find each exact value.

cos 195°

Use a half-angle identity to find each exact value.

sin 165°

Use the given information to find each of the following.

sin x/2 , given cos x = - 5/8, with π/2 < x < π