Textbook Question
Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2 .
Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 13
Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. csc θ , given that sin θ = ―3/7
Verified step by step guidance1
Recall the reciprocal identity relating cosecant and sine: \(\csc \theta = \frac{1}{\sin \theta}\).
Substitute the given value of \(\sin \theta = -\frac{3}{7}\) into the identity: \(\csc \theta = \frac{1}{-\frac{3}{7}}\).
Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\csc \theta = \frac{1}{-\frac{3}{7}} = -\frac{7}{3}\).
Check if the denominator is rationalized; since \(-\frac{7}{3}\) has no radical in the denominator, no further rationalization is needed.
Conclude that \(\csc \theta\) is the simplified reciprocal of \(\sin \theta\) based on the steps above.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Identities
Reciprocal identities relate trigonometric functions to each other by expressing one as the reciprocal of another. For example, cosecant (csc θ) is the reciprocal of sine (sin θ), so csc θ = 1/sin θ. This identity allows you to find csc θ directly when sin θ is known.
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Pythagorean Identities
Rationalizing the Denominator
Rationalizing the denominator involves eliminating any radicals or fractions from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression to simplify the expression and make it easier to interpret or use in further calculations.
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Evaluating Trigonometric Functions from Given Values
When given the value of one trigonometric function, you can find related functions using identities and algebraic manipulation. In this case, knowing sin θ allows you to find csc θ using the reciprocal identity, and you must consider the sign and domain of θ if needed.
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Related Practice
Textbook Question
Find the measure of each marked angle.
Textbook Question
Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2 .
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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. (―12 , ―5)
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Textbook Question
Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 45°
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Textbook Question
Find the measure of each marked angle.
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