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Chapter 7: Problem 88

Verify that each equation is an identity. $$\tan 2 \theta=\frac{-2 \tan \theta}{\sec ^{2} \theta-2}$$

Short Answer

Expert verified
The given equation is verified as an identity.

Step by step solution

01

Use Double-Angle Formulas

Start by using the double-angle formula for tangent: \ \(\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \). This formula will help us verify the given identity.
02

Express Secant in Terms of Tangent

Recall the Pythagorean identity: \ \(\text{sec}^2\theta = 1 + \tan^2\theta \). Substitute this into the given equation: \ \(\tan 2\theta = \frac{-2 \tan \theta}{\text{sec}^2 \theta - 2} \). \ \(\tan 2\theta = \frac{-2 \tan \theta}{1 + \tan^2 \theta - 2} \).
03

Simplify the Denominator

Simplify the denominator: \ \(\text{sec}^2\theta - 2 = 1 + \tan^2\theta - 2 = \tan^2\theta - 1 \). This gives us the equation: \ \(\tan 2 \theta = \frac{-2 \tan \theta}{\tan^2 \theta - 1} \).
04

Replace the Simplified Expression

Now replace the simplified expression back into the double-angle formula we used in Step 1: \ \(\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \). \ Observe that this is the negative reciprocal of the double-angle formula: \ \(\frac{-2 \tan \theta}{\tan^2 \theta - 1} = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
05

Conclude the Identity

Both expressions simplify to the same form: \ Therefore, the given equation \ \(\tan 2\theta = \frac{-2 \tan \theta}{\text{sec}^2\theta - 2} \) is verified to be an identity.

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Key Concepts

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

Double-Angle Formulas
Double-angle formulas are essential tools in trigonometry that allow us to express trigonometric functions of double angles (like \(2\theta\)) in terms of functions involving the single angle (like \(\theta\)). For tangent, the double-angle formula is: \[ \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \]. This formula is derived from the sum formulas for sine and cosine. Here's why they are useful:
  • They simplify complex trigonometric expressions.

  • They help in proving identities and solving trigonometric equations.
As we saw in the solution, starting with this double-angle formula lets us break down the problem into more manageable parts. By expressing \(\tan 2\theta\) using the double-angle formula, we make it easier to manipulate and simplify the given identity.
Pythagorean Identities
Pythagorean identities are another crucial set of tools in trigonometry, derived from the Pythagorean theorem. One of the most commonly used is: \[ \text{sec}^2\theta = 1 + \tan^2\theta \]. This identity let us replace \(\text{sec}^2\theta\) with \(1 + \tan^2\theta\), making the problem simpler to solve. Here鈥檚 how this helps:
  • It allows the introduction of trigonometric expressions involving squares into the equation.

  • Makes it easier to switch between different trigonometric functions.
For example, in our solution, substituting \(\text{sec}^2\theta\) with \(1 + \tan^2\theta\) helped us simplify the denominator, bringing us to the form \(\frac{-2\tan\theta}{\tan^2\theta - 1}\).
Tan Function
The tangent function, or \(\tan\), relates to both sine and cosine as \((\tan\theta = \frac{\sin\theta}{\cos\theta})\). Understanding the Tan function is key when working with identities like this. Here's why:
  • It is used in several trigonometric identities and formulas.

  • Tangent often appears in equations involving angles, making it a frequent target for simplification.
In the given identity, we work extensively with the tangent function. Recognizing that \(\tan 2\theta\) can be expressed in terms of \(\tan\theta\) helps verify the identity. Understanding how tangent interacts with other functions (like secant) opens the door to using both double-angle and Pythagorean identities to simplify and verify complex trigonometric expressions.

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Most popular questions from this chapter

Solve each problem. When a musical instrument creates a tone of \(110 \mathrm{Hz}\). it also creates tones at \(220,330,440,550,660, \ldots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. (Source: Benade, A.. Fundamentals of Musical Acoustics, Dover Publications.) (a) We can model this phenomenon using a graphing calculator. In the window \([0,0.03]\) by \([-1,1],\) graph the upper harmonics represented by the pressure $$ P=\frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t]+\frac{1}{4} \sin [2 \pi(440) t] $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above.

Verify that each trigonometric equation is an identity. $$(1+\sin x+\cos x)^{2}=2(1+\sin x)(1+\cos x)$$

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. $$\sin \left(\frac{\pi}{2}+\theta\right)$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(3 \theta=180^{\circ}, 630^{\circ}, 720^{\circ}, 930^{\circ} .\) What are the cor- responding values of \(\theta ?\)

Use the given information to find \(\cos (s+t)\) and \(\cos (s-t)\). \(\sin s=\frac{2}{3}\) and \(\sin t=-\frac{1}{3}, s\) in quadrant II and \(t\) in quadrant IV
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