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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 5.1.48
Capitolo 6, Problema 5.1.48

Perform each transformation. See Example 2.
Write cot x in terms of sin x.

Guida verificata passo dopo passo
1
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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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.
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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.
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Amplitude 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.
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Simplifying Trig Expressions
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For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

-tan x cos x


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

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Verify that each equation is an identity.

sin² θ (1 + cot² θ) - 1 = 0

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Find the remaining five trigonometric functions of θ.

cos θ = 1/5, θ in quadrant I

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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)

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Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

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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.

cos θ (cos θ - sec θ)

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