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

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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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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.

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Graph of Sine and Cosine Function

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.

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Graphs of Secant and Cosecant Functions

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.

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