Textbook Question
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.2
Textbook Question
Determine whether the positive or negative square root should be selected.
cos 58° = ±√ (1 + cos 116°)/2]
Verified step by step guidance1
Step 1: Understand the context of the problem. We are dealing with the cosine of an angle, and we need to determine whether to use the positive or negative square root in the expression \( \cos 58^\circ = \pm \sqrt{\frac{1 + \cos 116^\circ}{2}} \).
Step 2: Recall the cosine of a double angle identity: \( \cos 2\theta = 2\cos^2 \theta - 1 \). This can be rearranged to find \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \).
Step 3: Recognize that the expression \( \sqrt{\frac{1 + \cos 116^\circ}{2}} \) is derived from the identity for \( \cos^2 \theta \), where \( \theta = 58^\circ \) and \( 2\theta = 116^\circ \).
Step 4: Determine the sign of \( \cos 58^\circ \). Since \( 58^\circ \) is in the first quadrant, where cosine values are positive, we should select the positive square root.
Step 5: Conclude that \( \cos 58^\circ = \sqrt{\frac{1 + \cos 116^\circ}{2}} \) should use the positive square root, as \( 58^\circ \) is in the first quadrant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Function
The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic and oscillates between -1 and 1. Understanding the properties of the cosine function is essential for solving trigonometric equations and determining the values of angles.
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the cosine double angle formula, which states that cos(2θ) = 2cos²(θ) - 1. This identity can be used to simplify expressions and solve equations involving cosine, such as the one presented in the question.
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Square Roots and Their Sign
When dealing with square roots in trigonometric equations, it is crucial to consider both the positive and negative roots. The choice between the positive or negative root often depends on the context of the problem, such as the quadrant in which the angle lies. In this case, understanding the range of the cosine function and the specific angle values will guide the correct selection of the square root.
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