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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblem 2.3.27
Chapter 3, Problem 2.3.27

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°)

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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