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Problem 42
Simplify each expression.
±√[(1 - cos 5A)/(1 + cos 5A)]
Problem 44
Simplify each expression.
±√[(1 - cos (3θ/5))/2]
Problem 46
Verify that each equation is an identity.
cot² (x/2) = (1 + cos x)²/(sin² x)
Problem 48
Verify that each equation is an identity.
(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)
Problem 50
Verify that each equation is an identity.
tan (θ/2) = csc θ - cot θ
Problem 52
Verify that each equation is an identity.
cos x = (1 - tan² (x/2))/(1 + tan² (x/2))
Problem 65
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
cos 18°
Problem 67
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
cot 18°
Problem 72
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
tan 72°
Problem 74
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
csc 72°
Problem 76
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
sin 162°
Problem 5.RE.48
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
Problem 5.RE.32a
Use the given information to find sin(x + y).
cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III
Problem 5.RE..6
For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² x = ____
II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x
Problem 5.RE.40
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
Problem 5.RE.10
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc θ - sin θ
Problem 5.RE.32d
Use the given information to find the quadrant of x + y.
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
Problem 5.RE.20
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
-sin 35°
Problem 5.RE.8
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)
Problem 5.RE.30d
Use the given information to find the quadrant of x + y.
sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in quadrant III
Problem 5.RE.30b
Use the given information to find cos(x - y).
sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III
Problem 5.RE.44
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
Problem 5.RE.18
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
sin 35°
Problem 5.RE.16
Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using
a. difference identities
b. half-angle identities.
Problem 5.RE.32c
Use the given information to find tan(x + y).
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
Problem 5.RE.42
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
Problem 5.RE.36
Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0
Problem 5.RE.26
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
cos(-55°)
Problem 5.RE.30a
Use the given information to find sin(x + y).
sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III
Problem 5.RE.46
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)