Textbook Question
Simplify each expression. See Example 4.
⅛ sin 29.5° cos 29.5°
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6. Trigonometric Identities and More Equations
Double Angle Identities
Problem 8
Textbook Question
Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = -12/13 and sin θ > 0
Verified step by step guidance1
Identify the given information: \( \cos \theta = -\frac{12}{13} \) and \( \sin \theta > 0 \). This tells us that \( \theta \) is in the second quadrant because cosine is negative and sine is positive there.
Use the Pythagorean identity to find \( \sin \theta \): \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = -\frac{12}{13} \) to get \( \sin^2 \theta = 1 - \left(-\frac{12}{13}\right)^2 \).
Calculate \( \sin \theta \) by taking the positive square root (since \( \sin \theta > 0 \)): \( \sin \theta = +\sqrt{1 - \left(-\frac{12}{13}\right)^2} \).
Use the double-angle formulas for sine and cosine: \( \sin 2\theta = 2 \sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
Substitute the values of \( \sin \theta \) and \( \cos \theta \) into the double-angle formulas to find \( \sin 2\theta \) and \( \cos 2\theta \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Formulas
Double-angle formulas express trigonometric functions of 2θ in terms of functions of θ. For sine, sin(2θ) = 2 sin θ cos θ, and for cosine, cos(2θ) = cos² θ - sin² θ or equivalently 2 cos² θ - 1 or 1 - 2 sin² θ. These formulas are essential to find sin(2θ) and cos(2θ) from known values of sin θ and cos θ.
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Determining the Sign of Trigonometric Functions
The sign of sine and cosine depends on the quadrant where the angle θ lies. Given cos θ = -12/13 and sin θ > 0, θ is in the second quadrant where sine is positive and cosine is negative. This information helps correctly determine the values of sin θ and cos θ and their signs.
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Pythagorean Identity
The Pythagorean identity states that sin² θ + cos² θ = 1. Knowing cos θ allows calculation of sin θ by rearranging to sin θ = ±√(1 - cos² θ). The sign is chosen based on the quadrant information, ensuring accurate values for sine and cosine.
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