Textbook Question
Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α
Ch. 5 - Trigonometric Identities
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 6, Problem 5.1.22
Find sinθ.
csc θ = -8/5
Verified step by step guidance1
Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). This means \(\sin \theta = \frac{1}{\csc \theta}\).
Substitute the given value of \(\csc \theta = -\frac{8}{5}\) into the equation: \(\sin \theta = \frac{1}{-\frac{8}{5}}\).
Simplify the fraction by multiplying numerator and denominator: \(\sin \theta = -\frac{5}{8}\).
Consider the sign of \(\sin \theta\) based on the quadrant where \(\theta\) lies. Since \(\csc \theta\) is negative, \(\sin \theta\) is also negative, which matches the simplified value.
Conclude that \(\sin \theta = -\frac{5}{8}\), consistent with the given \(\csc \theta\).
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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 θ) is the reciprocal of the sine function (sin θ), meaning csc θ = 1/sin θ. To find sin θ when given csc θ, you take the reciprocal of the given value.
Recommended video:
6:04
Introduction to Trigonometric Functions
Sign of Trigonometric Functions in Quadrants
The sign of sine and cosecant depends on the quadrant where the angle θ lies. Since csc θ = -8/5 is negative, sin θ must also be negative, indicating θ is in either the third or fourth quadrant.
Recommended video:
6:36Quadratic Formula
Simplifying and Interpreting Fractional Values
Given csc θ = -8/5, sin θ is the reciprocal, so sin θ = -5/8. Understanding how to simplify and interpret these fractional values is essential for accurate calculation and further problem-solving.
Recommended video:
4:02Solving Linear Equations with Fractions
Related Practice
Textbook Question
The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)
Textbook Question
Factor each trigonometric expression.
sec² θ - 1
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Textbook Question
Verify that each equation is an identity.
(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
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Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
-sec² (-θ) + sin² (-θ) + cos² (-θ)
Textbook Question
Verify that each equation is an identity.
(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ