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Chapter 6: Problem 41

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta}=\tan \alpha-\tan \beta $$

Short Answer

Expert verified
The original equation \(\frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta}=\tan \alpha - \tan \beta\) is indeed equivalent, which has been verified through a series of substitutions and simplifications of the left side of the equation. Therefore, the original equation holds true.

Step by step solution

01

Rewrite the Left side using sine and cosine

Express the sine of the difference as an elementwise product of sine and cosine, in order to generate an equivalent expression for the left side of the identity. The sine of a difference can be rewritten as \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\), therefore:\[\frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta} = \frac{\sin \alpha \cos \beta - \cos \alpha \sin \beta}{\cos \alpha \cos \beta}\]
02

Break up into separate fractions

Divide each term in the numerator by the denominator, which yields two separate fractions:\[\frac{\sin \alpha \cos \beta - \cos \alpha \sin \beta}{\cos \alpha \cos \beta} = \frac{\sin \alpha \cos \beta}{\cos \alpha \cos \beta} - \frac{\cos \alpha \sin \beta}{\cos \alpha \cos \beta}\]
03

Simplify the fractions

Simplify each of the fractions by canceling out terms:\[\frac{\sin \alpha \cos \beta}{\cos \alpha \cos \beta} - \frac{\cos \alpha \sin \beta}{\cos \alpha \cos \beta} = \frac{\sin \alpha}{\cos \alpha} - \frac{\sin \beta}{\cos \beta}\]
04

Apply the definition of tangent

Rewrite each fraction using the definition of tangent, where \(\tan x = \sin x / \cos x\). This gives:\[\frac{\sin \alpha}{\cos \alpha} - \frac{\sin \beta}{\cos \beta} = \tan \alpha - \tan \beta\]

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Key Concepts

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

Sine and Cosine
Understanding sine and cosine is crucial when exploring trigonometric identities. In right-angled triangles, the sine of an angle is the ratio between the length of the opposite side and the hypotenuse. Cosine is the ratio between the adjacent side and the hypotenuse.
These basic definitions provide the foundation for more complex concepts, such as expressing the sine of angle differences.
For example, the identity \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\) allows us to express a sine term in terms of others.
Understanding these identities helps to manipulate expressions involving angles.
Difference of Angles Identity
The difference of angles identity is fundamental in trigonometry, simplifying the evaluation of trigonometric functions involving two angles. It helps in rewriting trigonometric expressions, such as separating or factorizing complex terms.
The formula for the sine of a difference, \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\), expands a sine term into a manageable form.
  • The first component \( \sin \alpha \cos \beta \) represents the product of the sine of the first angle and the cosine of the second angle.
  • The second component \(- \cos \alpha \sin \beta \) is the product of the cosine of the first angle and the sine of the second angle, subtracted from the first.
Recognizing these terms helps simplify expressions, allowing us to manipulate and solve equations more efficiently.
Tangent Function
The tangent function relates to sine and cosine through the ratio \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
As seen in the exercise, rewriting fractions using this identity simplifies expressions, moving from trigonometric ratios to more direct forms.
  • In the expression \(\frac{\sin \alpha}{\cos \alpha}\), it can be simplified to \(\tan \alpha\).
  • Similarly, \(\frac{\sin \beta}{\cos \beta}\) can be rewritten as \(\tan \beta\).
Thus, when applying the tangent definition to each adjusted term within these identities, we can transform cumbersome expressions into a straightforward form. This process makes evaluating and solving trigonometric equations significantly more accessible, highlighting the elegance of trigonometric identities.

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Most popular questions from this chapter

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ 3 \tan ^{2} x-\tan x-2=0 $$

Describe a natural periodic phenomenon. Give an example of a question that can be answered by a trigonometric equation in the study of this phenomenon.

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Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \cos 1.2 x \cos 0.8 x-\sin 1.2 x \sin 0.8 x=\cos 2 x $$
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