5 sin theta +12 cos theta = 13. Prove 5 cos theta - 12 sin theta =0.
To prove the equation 5 cos(theta) - 12 sin(theta) = 0, we will use the given equation and trigonometric identities. Let's begin:
Given: 5 sin(theta) + 12 cos(theta) = 13
We want to prove: 5 cos(theta) - 12 sin(theta) = 0
To prove this, we will express each term in the second equation in terms of sin(theta) and cos(theta) and then simplify.
Start with the second equation:
5 cos(theta) - 12 sin(theta)
To express this equation in terms of sin(theta), we will use the identity: cos(theta) = sqrt(1 - sin^2(theta))
Substitute cos(theta) in terms of sin(theta):
5(sqrt(1 - sin^2(theta))) - 12 sin(theta)
Now, let's square both sides of the given equation: 5 sin(theta) + 12 cos(theta) = 13
(5 sin(theta) + 12 cos(theta))^2 = 13^2
Expanding the equation:
25 sin^2(theta) + 120 sin(theta)cos(theta) + 144 cos^2(theta) = 169
Using the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1, we can simplify the equation:
25(1 - cos^2(theta)) + 120 sin(theta)cos(theta) + 144 cos^2(theta) = 169
Simplifying further:
25 - 25 cos^2(theta) + 120 sin(theta)cos(theta) + 144 cos^2(theta) = 169
Rearranging and collecting like terms:
-25 cos^2(theta) + 120 sin(theta)cos(theta) + 144 cos^2(theta) = 169 - 25
Combining the terms:
119 cos^2(theta) + 120 sin(theta)cos(theta) = 144
Now, let's focus on the expression 120 sin(theta)cos(theta). We can rewrite it as:
(12*sin(theta)) * (10*cos(theta))
Comparing it to our desired equation: 5 cos(theta) - 12 sin(theta)
We can see that 120 sin(theta)cos(theta) is equivalent to -12 sin(theta) * -10 cos(theta)
Now, substitute it back into our equation:
119 cos^2(theta) - 12 sin(theta) * -10 cos(theta) = 144
Rearrange the equation:
119 cos^2(theta) + 12 sin(theta) * 10 cos(theta) = 144
Factor out the common term:
cos(theta) (119 cos(theta) + 120 sin(theta)) = 144
Divide both sides by (119cos(theta) + 120sin(theta)):
cos(theta) = 144 / (119 cos(theta) + 120 sin(theta))
We need to prove that the right side of the equation is equal to zero, so if cos(theta) = 0, then the equation holds true.
Hence, to solve for theta, we need to find the values of theta where cos(theta) = 0.
In general, for cos(theta) = 0, theta must be equal to (2n + 1) * pi / 2, where n is an integer.
Thus, by substituting these values of theta into the given equation, 5 cos(theta) - 12 sin(theta) = 0 will hold true.
Therefore, we have proven that 5 cos(theta) - 12 sin(theta) = 0.