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

Verified step by step guidance

1. Step 1: Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
2. Step 2: Substitute the given values into the identity: \( \left(\frac{3}{4}\right)^2 + \left(\frac{2}{3}\right)^2 = 1 \).
3. Step 3: Calculate \( \left(\frac{3}{4}\right)^2 = \frac{9}{16} \) and \( \left(\frac{2}{3}\right)^2 = \frac{4}{9} \).
4. Step 4: Find a common denominator for \( \frac{9}{16} \) and \( \frac{4}{9} \) to add them together.
5. Step 5: Check if the sum of \( \frac{9}{16} \) and \( \frac{4}{9} \) equals 1 to determine if such an angle \( \theta \) exists.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the square of the sine and cosine functions must satisfy the equation sin²(θ) + cos²(θ) = 1. This identity is fundamental in trigonometry as it relates the two primary trigonometric functions and helps verify the validity of given function values.

Range of Trigonometric Functions

The sine and cosine functions have specific ranges: sin(θ) ranges from -1 to 1, and cos(θ) also ranges from -1 to 1. Understanding these ranges is crucial when determining if certain function values are possible for a given angle θ, as any values outside these ranges are invalid.

Existence of Angles in Trigonometry

In trigonometry, the existence of an angle θ that satisfies given sine and cosine values can be determined by checking if the values adhere to the Pythagorean identity. If the values do not satisfy this identity, it indicates that no such angle exists, which is essential for solving problems involving trigonometric functions.