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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 1 - Trigonometric FunctionsProblem 24
Chapter 2, Problem 24

Find the six trigonometric function values for each angle. Rationalize denominators when applicable.

Verified step by step guidance
1
Identify the given angle and determine its position on the coordinate plane or its reference angle if not directly given. This will help in finding the sine, cosine, and tangent values.
Recall the definitions of the six trigonometric functions in terms of a right triangle or the unit circle: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), \(\cot \theta = \frac{1}{\tan \theta}\).
Calculate the sine, cosine, and tangent values using the given angle or its reference triangle sides. If the problem provides side lengths, use those directly; if it provides an angle, use known values or the unit circle.
Find the cosecant, secant, and cotangent by taking the reciprocal of sine, cosine, and tangent respectively. Remember to rationalize denominators if any of these reciprocal values have radicals in the denominator.
Express all six trigonometric function values clearly, ensuring denominators are rationalized by multiplying numerator and denominator by the conjugate or appropriate radical to eliminate radicals from the denominator.

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Key Concepts

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

Six Trigonometric Functions

The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. They relate the angles of a right triangle to the ratios of its sides, and each function has a reciprocal counterpart (e.g., sine and cosecant). Understanding these functions is essential for finding their values for any given angle.
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Introduction to Trigonometric Functions

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression, making the expression simpler and more standardized, which is often required in trigonometric answers.
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Rationalizing Denominators

Evaluating Trigonometric Functions for Given Angles

To find the values of trigonometric functions for specific angles, one can use the unit circle, special right triangles (30°-60°-90°, 45°-45°-90°), or trigonometric identities. This process involves substituting the angle into the function definitions and simplifying the results, often requiring knowledge of exact values.
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Related Practice
Textbook Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (0, ―3)

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Textbook Question

Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , given that cos θ = 3/2 . "

Textbook Question

Find the measure of each marked angle. See Example 2.

Textbook Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1.

sin θ , given that csc θ = 1.25

Textbook Question

Find the measure of each marked angle. See Example 2.

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Textbook Question

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 147° 12' , 30° 19'