5sin thita +12cos theta=13 then find the value of 5cos thita -12sin theta
First, the name of θ is "theta" not "thita"
(5sinθ + 12cosθ)^2 = 169
25sin^2θ + 120sinθcosθ + 144cos^2θ = 169
(5cosθ - 12sinθ)^2 = 25cos^2θ - 120sinθcosθ + 144 sin^2θ
= 25cos^2θ - (169-25sin^2θ-144cos^2θ) + 144sin^2θ
= 25(cos^2θ+sin^2θ) + 144(sin^2θ+cos^2θ) - 169
= 25+144-169
= 0
or,
5sinθ + 12cosθ
= 13(5/13 sinθ + 12/13 cosθ)
now let x be the angle such that sinx = 12/13 and cosx = 5/13
= 13sin(θ+x)
So, we have
13sin(x+θ) = 13
sin(x+θ) = 1
Now, we also have
5cosθ - 12sinθ = 13(5/13 cosθ - 12/13 sinθ)
= 13cos(x+θ)
since sin(x+θ) = 1, cos(x+θ) = 0 as above
To find the value of 5cos(theta) - 12sin(theta), we need to first square the equation 5sin(theta) + 12cos(theta) = 13 and square the expression 5cos(theta) - 12sin(theta).
Let's start by squaring the equation 5sin(theta) + 12cos(theta) = 13:
(5sin(theta) + 12cos(theta))^2 = 13^2
25sin^2(theta) + 2 * 5sin(theta) * 12cos(theta) + 144cos^2(theta) = 169
25sin^2(theta) + 120sin(theta)cos(theta) + 144cos^2(theta) = 169
Now, let's square the expression 5cos(theta) - 12sin(theta):
(5cos(theta) - 12sin(theta))^2 = (5cos(theta))^2 - 2 * 5cos(theta) * 12sin(theta) + (12sin(theta))^2
25cos^2(theta) - 2 * 5cos(theta) * 12sin(theta) + 144sin^2(theta)
25cos^2(theta) - 120cos(theta)sin(theta) + 144sin^2(theta)
Now we can substitute the values back into the equation:
25cos^2(theta) - 120cos(theta)sin(theta) + 144sin^2(theta) = 169
25sin^2(theta) + 120sin(theta)cos(theta) + 144cos^2(theta) = 169
Since both equations are equal to 169, we can conclude that
25cos^2(theta) - 120cos(theta)sin(theta) + 144sin^2(theta) = 169
which means that 5cos(theta) - 12sin(theta) = ± √169
5cos(theta) - 12sin(theta) = ± 13
So the value of 5cos(theta) - 12sin(theta) can be either 13 or -13.
To find the value of 5cos(theta) - 12sin(theta) given the equation 5sin(theta) + 12cos(theta) = 13, we can use the trigonometric identity cos(theta) = sqrt(1 - sin^2(theta)).
Let's solve the given equation:
5sin(theta) + 12cos(theta) = 13
Rearranging the terms, we have:
12cos(theta) = 13 - 5sin(theta)
Divide both sides of the equation by 12:
cos(theta) = (13 - 5sin(theta)) / 12
Now, substitute this value of cos(theta) in terms of sin(theta) into the expression we need to find:
5cos(theta) - 12sin(theta) = 5[(13 - 5sin(theta)) / 12] - 12sin(theta)
Simplify this expression:
= (65 - 25sin(theta)) / 12 - 12sin(theta)
Now, our task is to find the value of 5cos(theta) - 12sin(theta). Unfortunately, we cannot determine the exact value of sin(theta) without further information or the value of theta itself.
If you have the value of theta, substitute it into the equation to find the value of 5cos(theta) - 12sin(theta). Otherwise, if you have more information about the problem or additional equations, please provide them so that a solution can be found.