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Chapter 5: Problem 24

Verify each identity. $$\sin 2 \theta=\frac{2 \cot \theta}{1+\cot ^{2} \theta}$$

Short Answer

Expert verified
The identity \(\sin 2 \theta=\frac{2 \cot \theta}{1+\cot ^{2} \theta}\) holds true.

Step by step solution

01

Rewrite the function using basic identities

By using the double angle identity, the left-hand side of the equation, \(\sin(2\theta)\), can be re-written as \(2\sin\theta \cos\theta\).
02

Change cotangent to cosine and sine

We know that \(\cot \theta\) is equivalent to \(\frac{\cos \theta}{\sin \theta}\). So, the right-hand side of the equation, \(\frac{2 \cot \theta}{1+\cot ^{2} \theta}\), can be re-written as \(\frac{ 2\cos\theta }{\sin\theta } / \frac{1+ \cos^2\theta }{\sin^2\theta }\).
03

Simplification

After doing the division on the right-hand side, we get, \(\frac{2\cos\theta}{1+ \cos^2\theta}\). Because we have the identity \(1+ \cos^2\theta = \sin^2\theta\), the term simplifies to \(2\sin\theta\cos\theta\) which is same as the left-hand side.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double Angle Identity
Understanding the double angle identity is crucial to working with trigonometric equations. This powerful tool in trigonometry expresses a trigonometric function of twice an angle in terms of functions of the original angle. Specifically, the double angle identities are for sine, cosine, and tangent functions.

For sine, the double angle identity is given by: \[\sin(2\theta) = 2\sin(\theta)\cos(\theta)\].What it does is express the sine of two times an angle as the product of the sine and cosine of the original angle. This formula is particularly useful for simplification in trigonometric proofs, as seen in the exercise where the left-hand side is transformed using this identity.
Cotangent Function
The cotangent function, often shown as \(\cot\), is a lesser-known but equally important trigonometric function. It's defined as the reciprocal of the tangent function or, in terms of sine and cosine, as the ratio of cosine to sine:\[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\].In the context of our exercise, recognizing that cotangent can be expressed in terms of sine and cosine is fundamental for converting the original equation into a more workable form.
Trigonometric Simplification
Trigonometric simplification involves rewriting trigonometric expressions in more manageable forms or in terms of other trigonometric functions. This process might encompass factoring, expanding, or using trigonometric identities. In the example provided, simplification was key in converting the right-hand side of the equation to a familiar form.It's essential to remember foundational identities such as \(1 + \cos^2(\theta) = \sin^2(\theta)\) as they facilitate the simplification of complex trigonometric expressions. Notably, this identity is a variation of the Pythagorean identity, modified for the purpose of the exercise.
Trigonometric Proofs
Trigonometric proofs require showing that two sides of an equation are identical by strategically manipulating one or both sides using trigonometric identities. This might include using double angle identities, reciprocal identities, or Pythagorean identities. The goal is to demonstrate that two seemingly different trigonometric expressions are, in fact, equivalent.In solving the exercise, the proof involved taking the given right-hand side and transforming it until it matched the left side after applying the aforementioned simplifications and identities. Once both sides display the same trigonometric expression, the identity is verified.

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Most popular questions from this chapter

In Exercises \(147-151,\) use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin x+\sin 2 x+\sin 3 x=0$$

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\sin x=0.7392$$

Use the power-reducing formulas to rewrite \(\sin ^{6} x\) as an equivalent expression that does not contain powers of trigonometric functions greater than 1

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When using the half-angle formulas for trigonometric functions of \(\frac{\alpha}{2},\) I determine the sign based on the quadrant in which \(\alpha\) lies.

Determine the amplitude and period of \(y=3 \cos 2 \pi x\) Then graph the function for \(-4 \leq x \leq 4\) (Section 4.5, Example 5)
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