Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
1/csc(90°-51°)

검증된 단계별 안내

Recognize that the expression is \( \frac{1}{\csc(90^\circ - 51^\circ)} \). The first step is to simplify the angle inside the cosecant function: calculate \( 90^\circ - 51^\circ \).

Recall the co-function identity for cosecant: \( \csc(90^\circ - \theta) = \sec(\theta) \). Use this identity to rewrite \( \csc(90^\circ - 51^\circ) \) as \( \sec(51^\circ) \).

Substitute the simplified expression back into the original fraction: \( \frac{1}{\sec(51^\circ)} \).

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so \( \frac{1}{\sec(51^\circ)} = \cos(51^\circ) \).

Use a calculator to find \( \cos(51^\circ) \) and round the result to six decimal places.

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주요 개념

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Reciprocal Trigonometric Functions

The cosecant function (csc) is the reciprocal of the sine function, meaning csc(θ) = 1/sin(θ). Understanding this relationship allows simplification of expressions involving csc by converting them into sine functions, which are often easier to evaluate.

Co-Function Identity

The co-function identity states that csc(90° - θ) = sec(θ), linking complementary angles in trigonometry. This identity helps simplify expressions by transforming csc of a complementary angle into sec, which can then be evaluated or further simplified.

Using a Calculator for Trigonometric Values

Calculators can approximate trigonometric values to a desired decimal precision. After simplifying the expression, input the angle in degrees or radians as required, and round the result to six decimal places for accuracy and consistency.

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1. 1/csc(90°-51°)