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Chapter 3: Problem 6

Prove the given identity. $$ \tan 3 \theta=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta} $$

Short Answer

Expert verified
To prove \( \tan 3 \theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta} \), apply the tangent addition formula and simplify using the double-angle formula.

Step by step solution

01

- Recall the tangent addition formula

To prove the given identity, start by recalling the tangent addition formula: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
02

- Express \tan(3\theta) using the addition formula

Express \( \tan 3 \theta \) as \( \tan(2\theta + \theta) \). Applying the tangent addition formula: \[ \tan(2 \theta + \theta) = \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} \]
03

- Find \tan(2\theta)

Next, we need \( \tan 2 \theta \). Use the double-angle formula: \[ \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \]
04

- Substitute \tan(2\theta) into the main expression

Substitute \( \tan 2 \theta \) into the expression for \( \tan(2\theta + \theta) \): \[ \tan(2 \theta + \theta) = \frac{\frac{2 \tan \theta}{1 - \tan^2 \theta} + \tan \theta}{1 - \frac{2 \tan \theta}{1 - \tan^2 \theta} \tan \theta} \]
05

- Simplify the numerator

Simplify the numerator: \[ \frac{2 \tan \theta}{1 - \tan^2 \theta} + \tan \theta = \frac{2 \tan \theta + \tan \theta (1 - \tan^2 \theta)}{1 - \tan^2 \theta} = \frac{3 \tan \theta - \tan^3 \theta}{1 - \tan^2 \theta} \]
06

- Simplify the denominator

Simplify the denominator: \[ 1 - \frac{2 \tan \theta \tan \theta}{1 - \tan^2 \theta} = \frac{1 - \tan^2 \theta - 2 \tan^2 \theta}{1 - \tan^2 \theta} = \frac{1 - 3 \tan^2 \theta}{1 - \tan^2 \theta} \]
07

- Combine and simplify

Combine the simplified numerator and denominator: \[ \frac{\frac{3 \tan \theta - \tan^3 \theta}{1 - \tan^2 \theta}}{\frac{1 - 3 \tan^2 \theta}{1 - \tan^2 \theta}} = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta} \]This matches the given identity.

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Key Concepts

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

Tangent Addition Formula
The **Tangent Addition Formula** helps us find the tangent of the sum of two angles. It's particularly useful in trigonometry to simplify complex expressions and prove identities. The formula is: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
Notice that this formula combines tangents of two separate angles into a single fraction. This is helpful in expressing more complex angles, like multiple angles of \( \theta \).
To apply this formula correctly:
  • Identify each angle to use in the addition.
  • Substitute the values into the formula.
  • Simplify the resulting expression.
This allows you to handle operations with angles in a straightforward manner and is a key step in trigonometric proofs.
Double-Angle Formula
The **Double-Angle Formula** for tangent simplifies the tangent of double angles (2胃) into expressions involving only tangent(胃). The formula is: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)
This formula allows us to break down more complex angles into smaller, more manageable parts. To understand how to use it:
  • Recognize the need to express a double angle (2胃).
  • Plug in the tangent of the base angle (胃) into the formula.
  • Simplify the expression to find the desired tangent value.
This tool is crucial in problems where you need to combine or simplify trigonometric expressions involving multiple angles.
Simplifying Expressions
Simplifying expressions in trigonometry often requires combining several formulas and algebraic techniques. When simplifying, follow these steps:
1. **Substitute Known Formulas** - Use relevant identities, such as the tangent addition and double-angle formulas, to express complex angles in simpler terms.
2. **Combine Like Terms** - Use algebraic techniques to merge like terms in numerators and denominators.
3. **Factor and Cancel** - Factor common elements where possible and cancel them out to reduce the expression further.
For example, in the given exercise:
  • Substitute \( \tan 2 \theta \) and use the tangent addition formula to express a more complex term.
  • Simplify the numerators and denominators separately by combining like terms.
  • Factor out common elements to further simplify the expression.
Through these techniques, you can often transform a complex trigonometric expression into a simpler, more readable form.

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Most popular questions from this chapter

Prove the given identity. $$ \cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta $$

Prove the given identity. $$ \cos ^{4} \theta-\sin ^{4} \theta=1-2 \sin ^{2} \theta $$

Find the exact values of \(\sin (A+B)\), \(\cos (A+B)\), and tan \((A+B)\). \( \sin A=\frac{8}{17}, \cos A=\frac{15}{17}, \sin B=\frac{24}{25}\), \(\cos B=\frac{7}{25}\)

There is a more general form for the instantaneous power \(p(t)=v(t) i(t)\) in an electrical circuit than the one in Example 3.22. The voltage \(v(t)\) and current \(i(t)\) can be given by $$ \begin{array}{l} v(t)=V_{m} \cos (\omega t+\theta) \\ i(t)=I_{m} \cos (\omega t+\phi) \end{array} $$ where \(\theta\) is called the phase angle. Show that \(p(t)\) can be written as $$ p(t)=\frac{1}{2} V_{m} I_{m} \cos (\theta-\phi)+\frac{1}{2} V_{m} I_{m} \cos (2 \omega t+\theta+\phi) $$.

Prove the given identity. $$ \frac{\sin 2 \theta}{\sin \theta}-\frac{\cos 2 \theta}{\cos \theta}=\sec \theta $$
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