if 5sin theta + 12cos theta is equal to 13 find the value of tan theta

See:

http://www.jiskha.com/display.cgi?id=1371300101

5/12

To find the value of tan(theta), we need to rewrite the given equation in terms of sin(theta) and cos(theta).

Given equation:
5sin(theta) + 12cos(theta) = 13

To rewrite this equation in terms of sin(theta) and cos(theta), we can divide both sides of the equation by cos(theta) since tan(theta) = sin(theta) / cos(theta).

After dividing by cos(theta), we get:
5sin(theta)/cos(theta) + 12cos(theta)/cos(theta) = 13/cos(theta)

This can be simplified to:
5tan(theta) + 12 = 13/cos(theta)

Now, we need to find the value of cos(theta) to simplify further.

To do that, we can use the Pythagorean Identity:
sin²(theta) + cos²(theta) = 1

Rearranging this equation, we have:
cos(theta) = sqrt(1 - sin²(theta))

Substituting the given equation, we have:
cos(theta) = sqrt(1 - (5sin(theta)/13)²)

Now, we can substitute this value back into the previous equation:
5tan(theta) + 12 = 13/sqrt(1 - (5sin(theta)/13)²)

Finally, we can solve this equation for tan(theta) by rearranging it:
5tan(theta) = 13/sqrt(1 - (5sin(theta)/13)²) - 12

Dividing both sides by 5, we get:
tan(theta) = (13/sqrt(1 - (5sin(theta)/13)²) - 12)/5

So, the value of tan(theta) is given by the expression:
(13/sqrt(1 - (5sin(theta)/13)²) - 12)/5