SOLUTION: How do I rewrite y=12sinx+5cosx in the form of y=Asin (x+theta)

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Question 631091: How do I rewrite y=12sinx+5cosx in the form of y=Asin (x+theta)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First we should keep in mind of where we are headed. We want an expression of the form:
y+=+A%2Asin%28x%2Btheta%29
It can be very help to keep your goal in mind. In fact we sometimes want to try to work backwards from the end toward the beginning.

So the next to last equation will look like:
y+=+A%2A%28sin%28x%29cos%28theta%29+%2B+cos%28x%29sin%28theta%29%29
What we have is
y = 12*sin(x) + 5*cos(x)
To get to
y+=+A%2A%28sin%28x%29cos%28theta%29+%2B+cos%28x%29sin%28theta%29%29
we will need to factor out some number, A, and have the 12 and 5 turn into the cos and sin on some angle called theta.

So what is this "magic" number that we can factor out that will 12 and 5 will turn into a cos and sin respectively. To figure this out it may help to draw a diagram. Draw a right triangle. Chose one of the acute angles to be theta. Make the side adjacent to theta a 12 and the side opposite to theta a 5. Use the Pythagorean Theorem to find the hypotenuse. You should find that the hypotenuse is 13. 13 is our magic number! Let's factor out a 13 from the right side of
y = 12*sin(x) + 5*cos(x)
and see how this works:
y+=+13%28%2812%2F13%29sin%28x%29+%2B+cos%28x%29%2A%285%2F13%29%29
Take a moment to let this sink in. You are probably not used to factoring out a 13 from a 12 or a 5. As it turns out any number (except zero) can be factored out of any other number. To understand this better, think about using the Distributive Property to multiply this back out. Won't the 13's all cancel out?

And are 12/13 and 5/13 the cos and sin of some angle? Look back at the triangle you drew. The answer should clearly be yes. The only difficulty is: How do we express this angle. 12/13 and 5/13 are not special angle values. And since theta is not a special angle, any number we use to express theta will be just a decimal approximation. Instead of an approximation we could use inverse Trig functions to express theta. Either of the following could be used. (If you have trouble with these, look at the your triangle to help yourself understand.)
theta+=+cos%5E-1%2812%2F13%29
or
theta+=+sin%5E-1%285%2F13%29

So here's the solution, from start to finish without much of the commentary from above:
y = 12*sin(x) + 5*cos(x)
Find and factor out the "magic" number, 13:
y+=+13%28%2812%2F13%29sin%28x%29+%2B+%285%2F13%29cos%28x%29%29
Replace the 12/13 and the 5/13 as a cos and sin of cos%5E-1%2812%2F13%29:

Use the sin(p+q) formula with "p" being "x" and "q" being cos%5E-1%2812%2F13%29
y+=+13%28sin%28x%2Bcos%5E-1%2812%2F13%29%29%29