Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

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Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 69

Capitolo 2, Problema 69

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

Guida verificata passo dopo passo

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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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)).

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Inverse Cosine

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.

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Introduction to Trigonometric Functions

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)).

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