Sum and Difference of Sines and Cosines

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Trigonometry › Sum and Difference of Sines and Cosines

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Solve for the following using the correct identity:

$cos(45 + 30) = \frac{\sqrt{6} - 2\sqrt{2}}{4}$

CORRECT

$cos(45 + 30) = \sqrt{6} - 2\sqrt{2}$

$cos(45 + 30) = \frac{\sqrt{2} - 2\sqrt{6}}{4}$

$cos(45 + 30) = - 2\sqrt{2}$

Explanation

The correct identity to use for this kind of problem is

$cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$ . We will let $\alpha = 45 = \frac{\pi}{4}$ and $\beta = 30 = \frac{\pi}{6}$ .

$cos(45 + 30) = cos(\frac{\pi}{4})cos(\frac{\pi}{6}) - sin(\frac{\pi}{4})sin(\frac{\pi}{6})$

$cos(45 + 30) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2})$

$cos(45 + 30) = \frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}$

$cos(45 + 30) = \frac{\sqrt{6} - 2\sqrt{2}}{4}$

What is the correct formula for the sum of two sines: ?

$sin(x)+sin(y) = 2sin\frac{x+y}{2}sin\frac{x-y}{2}$

$sin(x)+sin(y) = 2cos\frac{x+y}{2}sin\frac{x-y}{2}$

$sin(x)+sin(y) = 2sin\frac{x+y}{2}cos\frac{x+y}{2}$

$sin(x) + sin(y) = 2sin\frac{x + y}{2}cos\frac{x-y}{2}$

CORRECT

Explanation

This is a known trigonometry identity. Whenever you are adding two sine functions, you can plug and into the formula to solve for this sum

Solve for the following given that $x = 2\pi$ . Use the formula for the sum of two sines.

$sin(\frac{\pi}{3} +x) + sin(\frac{\pi}{3} - x)$

$\sqrt{3}$

CORRECT

$\pi$

$\frac{1}{2}$

Explanation

We begin by considering our formula for the sum of two sines

$sin(x) + sin(y) = 2sin\frac{x + y}{2}cos\frac{x-y}{2}$

We will let $x = \frac{\pi}{3} + x$ and $y = \frac{\pi}{3} - x$ and plug these values into our formula.

$sin(\frac{\pi}{3} +x) + sin(\frac{\pi}{3} - x) = 2sin\frac{(\frac{\pi}{3} + x) + (\frac{\pi}{3} - x)}{2}cos\frac{(\frac{\pi}{3} + x) - (\frac{\pi}{3} - x)}{2}$

$sin(\frac{\pi}{3} + x) + sin(\frac{\pi}{3} - x) = 2sin\frac{\frac{2\pi}{3}}{2}cos\frac{2x}{2}$

$sin(\frac{\pi}{3} + x) + sin(\frac{\pi}{3} - x) = 2sin\frac{\pi}{3}cosx$

$sin(\frac{\pi}{3} + x) + sin(\frac{\pi}{3} - x) = 2(\frac{\sqrt{3}}{2})(1)$

$sin(\frac{\pi}{3} + x) + sin(\frac{\pi}{3} - x) = \sqrt{3}$

Which of the following is the correct to complete the following identity: $sin(\alpha - \beta) =$ ___?

$sin(\alpha)cos(\alpha) - cos(\beta)sin(\beta)$

$cos(\alpha)cos(\beta) - cos(\alpha)cos(\beta)$

$sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta)$

CORRECT

$sin(\alpha)sin(\beta) - sin(\alpha)sin(\beta)$

Explanation

This is a known trigonometry identity and has been proven to be true. It is often helpful to solve for the quantity within a cosine function when there are unknowns or if the quantity needs to be simplified

Solve for the following using the correct identity:

$sin(\frac{\pi}{4} + \frac{\pi}{6})$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\sqrt{6} + 2\sqrt{2}$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\frac{\sqrt{2}}{2}$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\frac{\sqrt{6} + 2\sqrt{2}}{4}$

CORRECT

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\frac{\sqrt{6}}{4}$

Explanation

To solve this problem we must use the identity

$sin(\alpha + \beta) = sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta)$ . We will let $\alpha = \frac{\pi}{4}$ and $\beta = \frac{\pi}{6}$ .

$sin(\frac{\pi}{4} + \frac{\pi}{6}) = sin(\frac{\pi}{4})cos(\frac{\pi}{6}) + cos(\frac{\pi}{4})sin(\frac{\pi}{6})$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2}$

$sin(\frac{\pi}{4} + \frac{\pi}{6}) =\frac{\sqrt{6} + 2\sqrt{2}}{4}$

True or False: To solve for a problem in the form of $cos(\alpha - \beta)$ , I use the identity $cos(\alpha) - cos(\beta) = -2sin\frac{\alpha + \beta}{2}sin\frac{\alpha - \beta}{2}$ .