Concept Check Does there exist an angle θ with the function values cos θ = ⅔ and sin θ = ¾?

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Recall the Pythagorean identity for sine and cosine: \(\sin^2 \theta + \cos^2 \theta = 1\).

Substitute the given values into the identity: \(\left(\frac{3}{4}\right)^2 + \left(\frac{2}{3}\right)^2\).

Calculate each square separately: \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\) and \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\).

Add the two fractions: \(\frac{9}{16} + \frac{4}{9}\), and find a common denominator to combine them.

Compare the sum to 1; if the sum equals 1, then such an angle \(\theta\) exists, otherwise it does not.

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Pythagorean Identity

The Pythagorean identity states that for any angle θ, the square of the sine plus the square of the cosine equals one: sin²θ + cos²θ = 1. This fundamental relationship helps verify if given sine and cosine values correspond to a valid angle.

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Pythagorean Identities

Range of Sine and Cosine Functions

Sine and cosine functions have values that always lie between -1 and 1 inclusive. Any value outside this range is not possible for these trigonometric functions, which is essential when checking the validity of given values.

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Graph of Sine and Cosine Function

Existence of an Angle Given Sine and Cosine Values

To determine if an angle θ exists with specific sine and cosine values, both values must satisfy the Pythagorean identity and lie within the valid range. If these conditions hold, such an angle exists; otherwise, it does not.

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