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Problem 5.40
Simplify each expression. See Example 4.
1 - 2 sin² 22 ½°
Problem 5.4
For each expression in Column I, choose the expression from Column II that completes an identity.
4. cot x = ____
Problem 5.4
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
Problem 5.40
Simplify each expression.
±√[(1 + cos 20α)/2]
Problem 5.40
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan (π/4 + x)
Problem 5.40
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1
Problem 5.41
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
sec x/csc x
Problem 5.41
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
Problem 5.42
Simplify each expression. See Example 4.
cos² π/8 - 1/2
Problem 5.42
Simplify each expression.
±√[(1 - cos 5A)/(1 + cos 5A)]
Problem 5.42
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
Problem 5.42
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α
Problem 5.43
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
cos² x
Problem 5.43
Simplify each expression.
± √[(1 + cos (x/4))/2]
Problem 5.44
Simplify each expression. See Example 4.
tan 34°/2(1 - tan² 34°)
Problem 5.44
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 5.44
Simplify each expression.
±√[(1 - cos (3θ/5))/2]
Problem 5.44
Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(90° - θ)
Problem 5.44
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)
Problem 5.44
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
Problem 5.46
Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(θ - 270°)
Problem 5.46
Simplify each expression. See Example 4.
⅛ sin 29.5° cos 29.5°
Problem 5.46
Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.
Problem 5.46
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)
Problem 5.46
Verify that each equation is an identity.
cot² (x/2) = (1 + cos x)²/(sin² x)
Problem 5.46
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin(π + x)
Problem 5.46
Verify that each equation is an identity.
tan α/sec α = sin α
Problem 5.48
Simplify each expression. See Example 4.
cos² 2x - sin² 2x
Problem 5.48
Perform each transformation. See Example 2.
Write cot x in terms of sin x.
Problem 5.48
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