In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

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Identify the sides of the right triangle relative to angle \( \theta \) at vertex Q. The side opposite \( \theta \) is PR, the side adjacent to \( \theta \) is QR, and the hypotenuse is PQ.

Use the Pythagorean Theorem to find the missing side PR. The theorem states: \( \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \). Substitute the known values: \( 10^2 = PR^2 + 6^2 \).

Rearrange the equation to solve for PR: \( PR^2 = 10^2 - 6^2 \). Then take the square root of both sides to find PR: \( PR = \sqrt{10^2 - 6^2} \).

Once you have the length of PR, calculate the six trigonometric functions of \( \theta \) using the definitions: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \).

Substitute the side lengths into these formulas to express each trigonometric function in terms of the triangle's sides.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. It is expressed as a² + b² = c². This theorem is essential for finding the missing side length when two sides are known.

Right Triangle Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—relate the angles of a right triangle to the ratios of its sides. For an angle θ, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. The reciprocal functions are cosecant, secant, and cotangent.

Identifying Sides Relative to an Angle

In a right triangle, the side opposite the right angle is the hypotenuse. For a given angle θ, the side directly opposite is the opposite side, and the side next to θ (but not the hypotenuse) is the adjacent side. Correctly identifying these sides is crucial for applying trigonometric functions accurately.