Textbook Question
Use identities to correctly complete each sentence.
If sin θ = ⅔, then -sin(-θ) = ________________.
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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 5.6.4
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
Verified step by step guidance1
Recall the half-angle identity for sine: \(\sin \theta = \pm \sqrt{\frac{1 - \cos 2\theta}{2}}\). In this problem, \(\theta = -10^\circ\), so \(2\theta = -20^\circ\).
The expression given is \(\sin(-10^\circ) = \pm \sqrt{\frac{1 - \cos(-20^\circ)}{2}}\), which matches the half-angle formula. Our goal is to determine whether to use the positive or negative root.
Consider the sign of \(\sin(-10^\circ)\). Since sine is an odd function, \(\sin(-\alpha) = -\sin(\alpha)\), so \(\sin(-10^\circ) = -\sin(10^\circ)\).
Since \(\sin(10^\circ)\) is positive (because \(10^\circ\) is in the first quadrant where sine is positive), \(\sin(-10^\circ)\) must be negative.
Therefore, because \(\sin(-10^\circ)\) is negative, we select the negative square root in the half-angle formula: \(\sin(-10^\circ) = - \sqrt{\frac{1 - \cos(-20^\circ)}{2}}\).
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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 Sine
The half-angle identity expresses sine of an angle as ±√[(1 - cos(2θ))/2]. It helps find sin(θ) using the cosine of double the angle. The ± sign indicates that the sine value can be positive or negative depending on the angle's quadrant.
Recommended video:
05:06
Double Angle Identities
Sign of Sine Function in Different Quadrants
The sign of sine depends on the angle's quadrant on the unit circle. For negative angles, the position is measured clockwise from the positive x-axis. Since -10° lies in the fourth quadrant, where sine values are negative, the negative root should be chosen.
Recommended video:
06:14Sum and Difference of Sine & Cosine
Relationship Between Negative Angles and Trigonometric Functions
Sine is an odd function, meaning sin(-θ) = -sin(θ). This property helps determine the sign of sine for negative angles by relating it to the positive angle's sine value, confirming the sign choice in the half-angle formula.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²
Textbook Question
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 18°
Textbook Question
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
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.
tan(-θ)/sec θ
Textbook Question
Find the remaining five trigonometric functions of θ.
sin θ = -4/5, cos θ < 0