In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)

검증된 단계별 안내

Recognize that the expression is \( \sin(\cos^{-1}(\frac{\sqrt{2}}{2})) \). Here, \( \cos^{-1} \) is the inverse cosine function, which gives an angle \( \theta \) such that \( \cos \theta = \frac{\sqrt{2}}{2} \).

Let \( \theta = \cos^{-1}(\frac{\sqrt{2}}{2}) \). This means \( \cos \theta = \frac{\sqrt{2}}{2} \). Our goal is to find \( \sin \theta \).

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = \frac{\sqrt{2}}{2} \) into the identity to find \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left( \frac{\sqrt{2}}{2} \right)^2 \]

Simplify the expression inside the square root to find \( \sin^2 \theta \), then take the square root to find \( \sin \theta \): \[ \sin \theta = \pm \sqrt{1 - \left( \frac{\sqrt{2}}{2} \right)^2} \]

Determine the correct sign of \( \sin \theta \) by considering the range of \( \cos^{-1} \), which is \( [0, \pi] \). Since \( \theta \) is in the first or second quadrant, \( \sin \theta \) is non-negative in the first quadrant and positive in the second quadrant. Use this to select the positive or negative root.

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주요 개념

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Inverse Cosine Function (cos⁻¹)

The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x, typically within the range 0 to π radians. It helps find an angle when the cosine value is known, which is essential for evaluating expressions like sin(cos⁻¹(x)).

Right Triangle Interpretation of Trigonometric Functions

Trigonometric functions can be interpreted using right triangles, where cosine represents the adjacent side over hypotenuse. Using cos⁻¹(√2/2) gives an angle whose adjacent side and hypotenuse lengths can be used to find the opposite side, aiding in calculating sine of that angle.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This identity allows us to find sin(θ) when cos(θ) is known by rearranging to sin(θ) = √(1 - cos²θ), which is crucial for evaluating sin(cos⁻¹(√2/2)).