Textbook Question
Perform each transformation. See Example 2.
Write sec x in terms of sin x.
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.6.2
Determine whether the positive or negative square root should be selected.
cos 58° = ±√ (1 + cos 116°)/2]
Verified step by step guidance1
Recognize that the expression given is derived from the half-angle identity for cosine: \(\cos \theta = \pm \sqrt{\frac{1 + \cos 2\theta}{2}}\). Here, \(\theta = 58^\circ\) and \(2\theta = 116^\circ\).
Understand that the \(\pm\) sign depends on the quadrant in which the angle \(\theta\) lies. Since \(\theta = 58^\circ\) is in the first quadrant (0° to 90°), where cosine values are positive, the positive root should be selected.
Confirm the quadrant by recalling that cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
Therefore, for \(\cos 58^\circ\), choose the positive square root in the half-angle formula.
Summarize: Use the positive root because \(58^\circ\) is in the first quadrant where cosine is positive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Half-Angle Identity for Cosine
The half-angle identity expresses cosine of half an angle in terms of the cosine of the full angle: cos(θ/2) = ±√[(1 + cos θ)/2]. This formula helps find cosine values for angles that are half of known angles, but requires choosing the correct sign based on the angle's quadrant.
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05:06
Double Angle Identities
Determining the Sign of Trigonometric Functions
The sign of cosine depends on the quadrant in which the angle lies. Cosine is positive in the first and fourth quadrants and negative in the second and third. Identifying the quadrant of the half-angle (here 58°) is essential to select the correct positive or negative root.
Recommended video:
6:04Introduction to Trigonometric Functions
Relationship Between Angles and Their Cosine Values
Understanding how cosine values relate for angles and their multiples or halves is crucial. For example, cos 58° relates to cos 116° through the half-angle formula. Recognizing these relationships allows simplification and correct evaluation of trigonometric expressions.
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04:33Find the Angle Between Vectors
Related Practice
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
1 + cot(-θ)/cot(-θ)
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ - sin θ
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Textbook Question
Verify that each equation is an identity.
tan α/sec α = sin α
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot α sin α
Textbook Question
Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.