Multiple Choice
Identify what angle, , satisfies the following conditions.
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12. Trigonometric Functions
Trigonometric Functions on the Unit Circle
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For each expression, identify which coterminal angle to use & determine the exact value of the expression.
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Verified step by step guidance1
Step 1: Understand the problem. The goal is to evaluate the trigonometric expression \( \cos\left(-\frac{10\pi}{4}\right) \). To do this, we need to simplify the angle \( -\frac{10\pi}{4} \) by finding a coterminal angle within the interval \([0, 2\pi)\).
Step 2: Simplify the given angle. Start by reducing \( -\frac{10\pi}{4} \) to its simplest form. Divide the numerator and denominator by their greatest common divisor (2), resulting in \( -\frac{5\pi}{2} \).
Step 3: Find a coterminal angle. Since the angle is negative, add \( 2\pi \) repeatedly until the angle lies within \([0, 2\pi)\). Adding \( 2\pi \) to \( -\frac{5\pi}{2} \) gives \( -\frac{5\pi}{2} + 2\pi = \frac{-5\pi + 4\pi}{2} = -\frac{\pi}{2} \). Add \( 2\pi \) again to \( -\frac{\pi}{2} \) to get \( -\frac{\pi}{2} + 2\pi = \frac{-\pi + 4\pi}{2} = \frac{3\pi}{2} \).
Step 4: Evaluate the cosine function. The coterminal angle \( \frac{3\pi}{2} \) corresponds to the angle at the bottom of the unit circle (270 degrees). At this position, the cosine value is \( -1 \).
Step 5: Conclude the solution. The exact value of \( \cos\left(-\frac{10\pi}{4}\right) \) is \( -1 \).
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