In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.sin 37° csc 37°

검증된 단계별 안내

Recognize that \( \csc \theta \) is the reciprocal of \( \sin \theta \), which means \( \csc \theta = \frac{1}{\sin \theta} \).

Substitute \( \csc 37^\circ \) with \( \frac{1}{\sin 37^\circ} \) in the expression \( \sin 37^\circ \cdot \csc 37^\circ \).

The expression becomes \( \sin 37^\circ \cdot \frac{1}{\sin 37^\circ} \).

Simplify the expression by canceling \( \sin 37^\circ \) in the numerator and the denominator.

Conclude that the simplified expression equals 1.

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주요 개념

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

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For example, sin(37°) represents this ratio for a triangle with a 37-degree angle.

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). Therefore, csc(37°) is equal to the reciprocal of sin(37°), which means it represents the ratio of the hypotenuse to the opposite side in a right triangle.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this case, using the identity sin(θ) * csc(θ) = 1 can simplify the expression sin(37°) csc(37°) to 1, illustrating the relationship between sine and cosecant.