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Chapter 3: Problem 83

Complete the sum and difference identities for sine. a. \(\sin (x+y)= \)______ b. \(\sin (x-y)=\) ______

Short Answer

Expert verified
a) \( \sin x \cos y + \cos x \sin y \); b) \( \sin x \cos y - \cos x \sin y \)

Step by step solution

01

Identify the sum formula for sine

The sum formula for sine is: \( \sin (a + b) = \sin a \cos b + \cos a \sin b \).
02

Apply the sum formula to \( \sin (x + y) \)

Using the sum formula for sine, we get: \( \sin (x + y) = \sin x \cos y + \cos x \sin y \).
03

Identify the difference formula for sine

The difference formula for sine is: \( \sin (a - b) = \sin a \cos b - \cos a \sin b \).
04

Apply the difference formula to \( \sin (x - y) \)

Using the difference formula for sine, we get: \( \sin (x - y) = \sin x \cos y - \cos x \sin y \).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. They are fundamental tools in trigonometry and help simplify complex trigonometric expressions. Some of the most essential identities include identities for sine, cosine, and tangent. These identities are key in solving various trigonometric equations and simplifying expressions in both algebra and calculus.
Understanding these identities will help you more easily solve exercises and understand more advanced mathematical concepts.
Sum Formula for Sine
The sum formula for sine is an identity that expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles. The formula is given by:
\( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
This formula is extremely useful when you need to find the sine of an angle that can be expressed as the sum of two simpler angles.
  • Step 1: Identify the formula.
  • Step 2: Apply the formula by substituting the given angles in place of 'a' and 'b'.
    For example, for \sin(x + y)\, replace 'a' with 'x' and 'b' with 'y':\( \sin(x + y) = \sin x\cos y + \cos x\sin y \)Using this identity, complex calculations become more manageable.
Difference Formula for Sine
The difference formula for sine is another handy trigonometric identity. It allows you to find the sine of the difference between two angles based on the sines and cosines of the individual angles. The formula is:
\( \sin(a - b) = \sin a\cos b - \cos a\sin b \)
This is useful for angles that can be broken down into differences of simpler angles.
  • Step 1: Remember the formula.
  • Step 2: Substitute the given angles into the formula in place of 'a' and 'b'.
    For example, for \sin(x - y)\, replace 'a' with 'x' and 'b' with 'y':
    \( \sin(x - y) = \sin x\cos y - \cos x\sin y \)The difference formula simplifies the process of finding the sine of angle differences.
Step by Step Solutions
When solving trigonometric problems, breaking down the process into clear, manageable steps can make it much easier. Let's look at an example using our sum and difference identities for sine:
  • Step 1: Identify the sum formula for sine. The formula is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
  • Step 2: Apply the sum formula. For \sin(x + y)\, replace 'a' with 'x' and 'b' with 'y':
    \( \sin(x + y) = \sin x\cos y + \cos x\sin y \)
  • Step 3: Identify the difference formula for sine. The formula is \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
  • Step 4: Apply the difference formula. For \sin(x - y)\, replace 'a' with 'x' and 'b' with 'y':
    \( \sin(x - y) = \sin x\cos y - \cos x\sin y \)Following these steps will help you solve similar problems easily and accurately.

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Most popular questions from this chapter

Write each expression as a function of \(\alpha\) alone. $$ \cos (2 \pi-\alpha) $$

Evaluate each trigonometric function if possible. a. \(\sin (5 \pi / 6)\) b. \(\cos (\pi)\) c. \(\tan (3 \pi / 2)\) d. \(\sec (-\pi / 3)\) e. \(\csc (-\pi / 2)\) f. \(\cot (5 \pi / 4)\)

Verify that each equation is an identity. \(\frac{\sin (x+y)}{\sin x \cos y}=1+\cot x \tan y\)

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\). \(\cos 2 \alpha=1 / 3\) and \(360^{\circ}<2 \alpha<450^{\circ}\)

Prove that each equation is an identity: $$ \sin ^{2} x-\sin ^{2} y=\sin (x+y) \sin (x-y) $$
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