Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.14

Chapter 3, Problem 2.3.14

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. csc 145° 45'

Verified step by step guidance

Convert the angle from degrees and minutes to decimal degrees. Since 1 minute is \( \frac{1}{60} \) of a degree, calculate \( 45' = \frac{45}{60} = 0.75 \) degrees. Then, add this to 145 degrees to get the total angle: \( 145 + 0.75 = 145.75^\circ \).

Recall that \( \csc \theta = \frac{1}{\sin \theta} \). So, to find \( \csc 145^\circ 45' \), you first need to find \( \sin 145.75^\circ \).

Use a calculator set to degree mode to find \( \sin 145.75^\circ \).

Calculate the reciprocal of the sine value to find the cosecant: \( \csc 145.75^\circ = \frac{1}{\sin 145.75^\circ} \).

Round your final answer to six decimal places as requested.

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Key Concepts

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

Understanding the Cosecant Function

The cosecant function, csc(θ), is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). To find csc of an angle, first find the sine of that angle, then take its reciprocal. This is essential for evaluating expressions involving csc.

Converting Degrees and Minutes to Decimal Degrees

Angles given in degrees and minutes must be converted to decimal degrees before using a calculator. Since 1 minute equals 1/60 of a degree, convert by adding minutes divided by 60 to the degrees. For example, 145° 45' = 145 + 45/60 = 145.75°.

Using a Calculator to Approximate Trigonometric Values

After simplifying and converting the angle, use a scientific calculator set to degree mode to find the sine value. Then compute the reciprocal for cosecant. Round the final answer to six decimal places as required for precision.

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