In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅

검증된 단계별 안내

Identify the given information: \( \sec \alpha = -3 \) and the interval \( \frac{\pi}{2} < \alpha < \pi \). Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \).

Find \( \cos \alpha \) by taking the reciprocal of \( \sec \alpha \): \( \cos \alpha = \frac{1}{\sec \alpha} = \frac{1}{-3} = -\frac{1}{3} \).

Since \( \alpha \) is in the interval \( \frac{\pi}{2} < \alpha < \pi \), which corresponds to the second quadrant, note that cosine values are negative there, confirming the sign of \( \cos \alpha \).

Use the double-angle formula for cosine to find \( \cos 2\alpha \): \[ \cos 2\alpha = 2 \cos^2 \alpha - 1 \].

Substitute \( \cos \alpha = -\frac{1}{3} \) into the double-angle formula and simplify the expression to find the exact value of \( \cos 2\alpha \).

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주요 개념

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Secant and Cosine Relationship

Secant (sec) is the reciprocal of cosine (cos), so sec α = 1/cos α. Given sec α = -3, we find cos α by taking the reciprocal, resulting in cos α = -1/3. Understanding this reciprocal relationship is essential to convert between secant and cosine values.

Angle Interval and Sign of Trigonometric Functions

The interval π/2 < α < π places α in the second quadrant, where cosine values are negative. This information confirms the sign of cos α, ensuring the correct value is chosen when solving for cosine or related functions.

Half-Angle Formulas

To find cos(α/2), use the half-angle identity: cos(α/2) = ±√[(1 + cos α)/2]. The sign depends on the quadrant of α/2. Applying this formula allows calculation of the exact value of cos(α/2) from the known cos α.

In Exercises 55–58, use the given information to find the exact value of each of the following: b. cos(α/2)sec α = ﹣3, 𝝅/2 < α < 𝝅