Trigonometry - Complete a Proof Using Sums, Differences, or Products of Sines and Cosines

Simplify by applying the compound angle formula:

Explanation

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that $\cos(x - y) = sinxsiny + cosxcosy$ and $\cos(x + y) = sinxsiny - cosxcosy$ , substitution yields the following:

$(1/2)(\cos(x - y) + \cos(x + y))$

This is the formula for the product of two cosines, $cosxcosy = (1/2)(\cos(x - y) + \cos(x + y))$ .

Which of the following correctly demonstrates the compound angle formula?

$\cos(x + y) = sinxcosy - cosxsiny$

$\cos(x + y) = sinxsiny - cosxcosy$

$\cos(x - y) = sinxsiny - cosxcosy$

$\cos(x - y) = sinxcosy - cosxsiny$

Explanation

The compound angle formula for cosines states that $\cos(x +- y) = sinxsiny -+ cosxcosy$ .

True or false:

$\sin x + \sin y = \sin (x + y)$ .

True

False

Cannot be determined

Explanation

The sum of sines is given by the formula $\sin x + \sin y = \2\sin ((x + y)/2) * \cos((x - y)/2)$ .

Which of the following correctly demonstrates the compound angle formula?

$\sin(x + y) = sinxcosy - cosxsiny$

$\sin(x + y) = sinxsiny - cosxcosy$

$\sin(x - y) = sinxsiny - cosxcosy$

$\sin(x - y) = sinxcosy - cosxsiny$

Explanation

The compound angle formula for sines states that $\sin(x +- y) = sinxcosy +- cosxsiny$ .

Using $\sin(90 - x) = \cos x$ and the formula for the sum of two sines, rewrite the sum of cosine and sine:

$\cos x + \sin x$

$\sin(45) * \cos(45 - x)$

$\sin(90) * \cos(90 - x)$

$2\sin(45) * \cos(45 - x)$

$2\sin(90) * \cos(90 - x)$

Explanation

Substitute $\sin(90 - x)$ for $\cos x$ :

$\cos x + \sin x$

$\sin(90 - x) + \sin x$

Apply the formula for the sum of two sines, $\sin x + \sin y = 2\sin ((x + y)/2) * \cos((x - y)/2)$ :

$\sin(90 - x) + \sin x$

$2\sin((90 - x + x)/2) * \cos((90 - x - x)/2)$

$2\sin(45) * \cos(45 - x)$

True or false: $\cos x - \cos y = \cos (x - y)$ .

True

False

Cannot be determined

Explanation

The difference of cosines is given by the formula $\cos x - \cos y = 2\sin ((x + y)/2) * \sin((y - x)/2)$ .

Simplify by applying the compound angle formula:

Explanation

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that $\sin (x - y) = sinxcosy - cosxsiny$ and $\sin (x + y) = sinxcosy + cosxsiny$ , substitution yields the following: