Textbook Question
Use the given information to find each of the following.
cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 30
Textbook Question
Use the given information to find each of the following.
sin x, given cos 2x = 2/3 , with π < x < 3π/2
Verified step by step guidance1
Recall the double-angle identity for cosine: \(\cos 2x = 1 - 2\sin^2 x\). This relates \(\cos 2x\) to \(\sin x\).
Substitute the given value \(\cos 2x = \frac{2}{3}\) into the identity: \(\frac{2}{3} = 1 - 2\sin^2 x\).
Rearrange the equation to solve for \(\sin^2 x\): \(2\sin^2 x = 1 - \frac{2}{3}\), then simplify the right side.
Calculate \(\sin^2 x\) by dividing both sides by 2: \(\sin^2 x = \frac{1 - \frac{2}{3}}{2}\).
Determine the sign of \(\sin x\) based on the interval \(\pi < x < \frac{3\pi}{2}\). Since \(x\) is in the third quadrant, where sine is negative, take the negative square root to find \(\sin x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Identity for Cosine
The double-angle identity states that cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows us to express cos(2x) in terms of sin(x) or cos(x), which is essential for finding sin(x) when cos(2x) is known.
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Determining the Sign of Trigonometric Functions Based on Quadrants
The value of sin(x) and cos(x) depends on the quadrant where angle x lies. Since π < x < 3π/2 places x in the third quadrant, both sine and cosine values are negative there. This information helps determine the correct sign of sin(x) after calculation.
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Using Pythagorean Identities to Relate Sine and Cosine
The Pythagorean identity sin²(x) + cos²(x) = 1 links sine and cosine values. After expressing cos(2x) in terms of sin(x) or cos(x), this identity helps solve for sin(x) by substituting and rearranging the equation.
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