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Chapter 3: Problem 46

Write each expression as a function of \(\alpha\) alone. \(\tan (\pi / 4-\alpha)\)

Short Answer

Expert verified
\ \tan(\pi / 4 - \alpha) = \frac{1 - \tan(\alpha)}{1 + \tan(\alpha)}

Step by step solution

01

Understand the Tangent Difference Formula

Recall the tangent difference formula: \ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)} In this exercise, \(a\) is \(\pi / 4\) and \(b\) is \(\alpha\).
02

Apply Known Tangent Values

We can substitute \(a = \pi / 4\) and \(b = \alpha\) into the formula. We also know that \(\tan(\pi / 4) = 1\): \tan(\pi / 4 - \alpha) = \frac{1 - \tan(\alpha)}{1 + \tan(\alpha)}
03

Simplify the Expression

The expression \(\frac{1 - \tan(\alpha)}{1 + \tan(\alpha)}\) is already in its simplest form as a function of \(\alpha\).

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Key Concepts

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

tangent difference formula
The tangent difference formula is a key trigonometric identity used to find the tangent of the difference between two angles. This formula is given as: \ \[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)} \].

It鈥檚 essential to apply this formula correctly to solve problems involving the subtraction of angles. Here, we identify values for the angles involved. Specifically, in our problem, we have:
\[ a = \frac{\pi}{4} \],
\[ b = \alpha \].

Understanding and memorizing the tangent difference formula can simplify complex trigonometric expressions, especially in problems that involve angle subtraction.
angle subtraction
Angle subtraction in trigonometry involves finding the trigonometric functions of the difference between two angles. By using known values of common angles, such as \( \pi / 4 \), angle subtraction becomes simpler.

In our given exercise, we are working with \( \pi / 4 \) and \( \alpha \). To solve \( \tan(\pi / 4 - \alpha) \), we substitute into the tangent difference formula we have discussed earlier.
Here鈥檚 the breakdown of the substitution:
  • \( \tan(a) = \tan(\pi / 4) \)
    It is known that \( \tan(\pi / 4) = 1 \)
  • \( \tan(b) = \tan(\alpha) \)

Substituting these values:
\[ \tan(\pi / 4 - \alpha) = \frac{1 - \tan(\alpha)}{1 + \tan(\alpha)} \]
This substitution simplifies our angle subtraction problem significantly.
trigonometric simplification
Trigonometric simplification refers to the process of breaking down complex trigonometric expressions into simpler forms. For our exercise, the original expression \( \tan(\pi / 4 - \alpha) \) has been simplified using the tangent difference formula to
\[ \frac{1 - \tan(\alpha)}{1 + \tan(\alpha)} \].

This step is crucial because it reduces the expression to a function solely involving \( \alpha \), making it easier to understand and use in further calculations.

Trigonometric simplification often involves the following steps:
  • Identify the appropriate trigonometric identity (like the tangent difference formula).
  • Substitute known values or expressions into the identity.
  • Perform algebraic operations to reduce the expression to its simplest form.

Practicing these simplification techniques can significantly improve your problem-solving skills in trigonometry.

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Most popular questions from this chapter

Find the exact value of \(\tan (x / 2)\) given that \(\sin (x)=-\sqrt{8 / 9}\) and \(\pi

Simplify each expression by applying the odd/even identities, cofunction identities, and cosine of a sum or difference identities. Do not use a calculator $$ \sin (\pi / 2-z) \cos (-z)-\cos (\pi / 2-z) \sin (-z) $$

Without using a calculator find the exact value of \(\sin \left(\frac{\pi}{24}\right) \cos \left(\frac{\pi}{24}\right) \cos \left(\frac{\pi}{12}\right) \cos \left(\frac{\pi}{6}\right) \cos \left(\frac{\pi}{3}\right)\)

Simplify each expression by applying the odd/even identities, cofunction identities, and cosine of a sum or difference identities. Do not use a calculator $$ \sin \left(85^{\circ}\right) \sin \left(40^{\circ}\right)+\sin \left(-5^{\circ}\right) \sin \left(-50^{\circ}\right) $$

Write each expression as a function of \(\alpha\) alone. $$ \cos \left(\alpha-360^{\circ}\right) $$
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