Textbook Question
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
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 5.1.48
Perform each transformation. See Example 2.
Write cot x in terms of sin x.
Verified step by step guidance1
Recall the definition of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\).
Since the problem asks to write \(\cot x\) in terms of \(\sin x\) only, we need to express \(\cos x\) in terms of \(\sin x\).
Use the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\), which can be rearranged to \(\cos x = \pm \sqrt{1 - \sin^2 x}\).
Substitute \(\cos x\) back into the cotangent expression: \(\cot x = \frac{\pm \sqrt{1 - \sin^2 x}}{\sin x}\).
Note that the sign depends on the quadrant where \(x\) lies, so the final expression for \(\cot x\) in terms of \(\sin x\) is \(\cot x = \pm \frac{\sqrt{1 - \sin^2 x}}{\sin x}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Cotangent
Cotangent (cot x) is the reciprocal of the tangent function. It can be expressed as cot x = cos x / sin x, which relates cotangent directly to sine and cosine functions.
Recommended video:
5:37
Introduction to Cotangent Graph
Relationship Between Sine and Cosine
Sine and cosine are fundamental trigonometric functions related by the Pythagorean identity: sin²x + cos²x = 1. This identity allows expressing one function in terms of the other, which is useful when rewriting cotangent solely in terms of sine.
Recommended video:
5:05Amplitude and Reflection of Sine and Cosine
Algebraic Manipulation of Trigonometric Expressions
Transforming trigonometric expressions often requires algebraic skills such as substitution and rearrangement. To write cot x in terms of sin x, one must manipulate the expression cot x = cos x / sin x using identities and algebraic steps.
Recommended video:
6:36Simplifying Trig Expressions
Related Practice
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot t tan t
Textbook Question
Factor each trigonometric expression.
sec² θ - 1
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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.
(sin θ - cos θ) (csc θ + sec θ)
6
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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
Match each expression in Column I with its value in Column II.
10. cos 67.5°
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