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Chapter 5: Problem 64

Prove the identity. $$\sin (x+y)+\sin (x-y)=2 \sin x \cos y$$

Short Answer

Expert verified
\The Identity: \(\sin (x+y)+\sin (x-y)=2 \sin x \cos y\) is proven to be true.

Step by step solution

01

Express sin(x+y) and sin(x-y) in terms of sines and cosines

Remember, by the sum and difference formulas for sine, we know that: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\) and \(\sin(x-y) = \sin x \cos y - \cos x \sin y\)
02

Add sin(x+y) and sin(x-y)

Adding these two formulas will elminate the last term of both: \(\sin(x+y) + \sin(x-y) = [\sin x \cos y + \cos x \sin y] + [\sin x \cos y - \cos x \sin y]\) This simplifies to: \(\sin(x+y) + \sin(x-y) = 2 \sin x \cos y\)
03

Write the Final Result

\(\sin (x+y)+\sin (x-y)=2 \sin x \cos y\)

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Key Concepts

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

Sum and Difference Formulas
Sum and difference formulas are key identities in trigonometry. These formulas help us express the sine and cosine of the sum or difference of two angles in terms of sines and cosines of those individual angles.
For instance, the sine sum formula is given by:
  • \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
Similarly, the sine difference formula is:
  • \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)
By adding these formulas, the terms \( \cos x \sin y \) cancel each other, simplifying the expression. These identities are extremely helpful in simplifying expressions involving trigonometric functions.
They not only provide a means to break down trigonometric expressions but also make them easier to solve or verify, such as proving identities.
Sine Function
The sine function is one of the principal functions in trigonometry. It is used to relate the angles of a triangle to the ratios of two of its sides. For an angle \( \theta \), the sine function is denoted as \( \sin \theta \).
This function outputs the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. Its value ranges from -1 to 1 as the angle varies.
In the context of the sum and difference formulas, the sine function is particularly significant as it helps determine how we combine angles:
  • For \( \sin(x+y) \), it combines the angles \( x \) and \( y \) through their respective sines and cosines.
  • The sine difference formula, \( \sin(x-y) \), similarly engages the complementary nature of angle difference in trigonometric calculations.
Having a solid grasp on sine helps when applying these formulas in problem-solving.
Cosine Function
The cosine function, similar to the sine function, plays a pivotal role in trigonometry. It is denoted as \( \cos \theta \) where \( \theta \) represents an angle. Like sine, cosine connects angles to side ratios in right triangles.
Specifically, it measures the length of the adjacent side over the hypotenuse. The range of the cosine function is also from -1 to 1, fluctuating as angles change.
When considering sum and difference formulas, the cosine function is crucial:
  • The formula \( \sin(x+y) = \sin x \cos y + \cos x \sin y \) shows how cosine interplays with sine in angle additions.
  • In \( \sin(x-y) = \sin x \cos y - \cos x \sin y \), cosine similarly defines the angle subtraction relationship.
Understanding the behavior of cosine not only aids in evaluating trigonometric expressions but also in visualizing how angle manipulations influence function outcomes.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x+1}{1-\tan x}$$

Write the trigonometric expression as an algebraic expression. $$\sin (\arctan 2 x-\arccos x)$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin 6 x+\sin 2 x=0$$

Prove the identity. $$\cos (\pi-\theta)+\sin \left(\frac{\pi}{2}+\theta\right)=0$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$
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