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Chapter 5: Problem 115

Rewrite the expression as a single logarithm and simplify the result. $$\ln |\sin x|+\ln |\cot x|$$

Short Answer

Expert verified
The rewritten and simplified expression is \(\ln (|\cos x|)\).

Step by step solution

01

Recognize the Functions and Properties

Recognize the functions involved and recall that the logarithms with the same base \(a\) (natural logarithm base \(e\) in this case) can be combined using properties of logarithms. Moreover, understand that the cotangent function is the reciprocal of the tangent function, such that \(\cot x = 1 / \tan x\).
02

Apply Logarithm Properties

Apply the property that the sum of two logarithms is equivalent to the logarithm of their product. Therefore, \(\ln |\sin x| + \ln |\cot x| = \ln (|\sin x| \cdot |\cot x|)\).
03

Simplify Expression

Simplify the expression inside the logarithm. Remember that cotangent is the reciprocal of tangent, so \(|\cot x| = |1 / \tan x|\). Further simplify to \(\ln (|\sin x| \cdot |1 / \tan x|) = \ln (|\sin x / \tan x|)\). Since \(\tan x = \sin x / \cos x\), it leads to the simplified expression \(\ln (|\cos x|)\).

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Most popular questions from this chapter

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}$$

Find the \(x\) -intercepts of the graph. $$y=\tan ^{2}\left(\frac{\pi x}{6}\right)-3$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\cos ^{2} x-\sin x$$ Trigonometric Equation $$-2 \sin x \cos x-\cos x=0$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sin ^{2} x+\cos x$$ Trigonometric Equation $$2 \sin x \cos x-\sin x=0$$

Fill in the blank. \(\cos (u-v)=\)_____
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