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Chapter 8: Problem 70

Establish each identity. \(\tan \theta+\tan \left(\theta+120^{\circ}\right)+\tan \left(\theta+240^{\circ}\right)=3 \tan (3 \theta)\)

Short Answer

Expert verified
The identity simplifies to \(3 \tan(3 \theta)\).

Step by step solution

01

Recognize the trigonometric identities

Use the angle addition identities for tangent:\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
02

Write down the angles involved

Here, the angles are \(\theta\), \(\theta + 120^{\circ}\), and \(\theta + 240^{\circ}\).
03

Apply the tangent of sum formula to \(\tan(\theta + 120^{\circ})\)

Using the addition formula:\[ \theta + 120^{\circ} \]\[ \tan(\theta + 120^{\circ}) = \frac{\tan \theta + \tan 120^{\circ}}{1 - \tan \theta \tan 120^{\circ}} = \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} \]
04

Apply the tangent of sum formula to \(\tan(\theta + 240^{\circ})\)

Using the addition formula:\[ \theta + 240^{\circ} \]\[ \tan(\theta + 240^{\circ}) = \frac{\tan \theta + \tan 240^{\circ}}{1 - \tan \theta \tan 240^{\circ}} = \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} \]
05

Add the three tangent functions together

Sum the three results:\[ \tan \theta + \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} + \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} \]
06

Simplify the expression

Combine the fractions and use algebraic manipulations to bring it to the form of \(3 \tan(3 \theta)\).

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Key Concepts

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

tangent of sum formula
When dealing with trigonometric identities, one essential formula to understand is the tangent of sum formula. It reads as follows:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
This formula allows us to compute the tangent of the sum of two angles.
In essence, it simplifies finding the tangent of non-standard angles by expressing them as sums of commonly known angles.Utilizing this relationship requires keeping track of both the numerators and the denominators in the fractional form.
Make sure to recognize that each part of the tangent formula embodies the mix of both angles we initially started with.
This formula can be handy in various trigonometric transformations and simplifications.
angle addition identities
Angle addition identities in trigonometry help us break down complex expressions involving sums of angles. They include formulas for sine, cosine, and tangent. The key for tangent is given as:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
In the given exercise:
\( a \) is \( \theta \) and \( b \) is \( 120^{\circ} \) or \( 240^{\circ} \).
We used these identities to manipulate trigonometric expressions. It is all about transforming the given angles into simpler forms. For the angles 120掳 and 240掳, the tangent values are special:
  • \( \tan 120^{\circ} = - \sqrt{3} \)
  • \( \tan 240^{\circ} = \sqrt{3} \)

When breaking the problem, remember that these identities link different trigonometric functions through algebra.
algebraic manipulation
Once the trigonometric identities are applied, the next step is algebraic manipulation. This involved adding fractions and simplifying complex expressions.
Here's the summarized manipulation:
  • Combine \( \tan \theta \) with \( \frac{ \tan \theta - \sqrt{3}}{ 1 + \tan \theta \sqrt{3}} \) and \( \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} \).
  • Work on finding a common denominator for these fractions.
  • Simplify using known identities and algebraic rules.

    • Once simplified, we can refine the expression to match \( 3 \tan (3 \theta) \). Algebraic manipulation isn't just about combining fractions, but also about simplifying expressions to match desired identities. Be patient, and break down the steps one by one, using fundamental properties of algebra.
    This incremental process will clarify how the expressions transform into more usable forms.

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Most popular questions from this chapter

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Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, \(\theta\) is the viewing angle. Suppose that you sit \(x\) feet from the screen. The viewing angle \(\theta\) is given by the function $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ (a) What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet? (b) If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle? (c) Using a graphing utility, graph $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ What value of \(x\) results in the largest viewing angle?
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