if 3 cot theta =4 find the value of 5cos theta-2sintheta/5cos theta+3sin theta
3 cot ( theta ) = 4 Divide both sides by 3
cot ( theta ) = 4 / 3
sin ( theta ) = + OR - 1 / sqrt [ 1 + cot ( teta ) ^ 2 ]
In this case :
sin ( theta ) = + OR - 1 / sqrt [ 1 + ( 4 / 3 ) ^ 2 ]
sin ( theta ) = + OR - 1 / sqrt ( 1 + 16 / 9 )
sin ( theta ) = + OR - 1 / sqrt ( 9 / 9 + 16 / 9 )
sin ( theta ) = + OR - 1 / sqrt ( 25 / 9 )
sin ( theta ) = + OR - 1 / ( 5 / 3 )
sin ( theta ) = + OR - 3 / 5
cos ( theta ) = + OR - cot ( theta ) / sqrt [ 1 + cot ( teta ) ^ 2 ]
In this case :
cos ( theta ) = + OR - ( 4 / 3 ) / sqrt [ 1 + ( 4 / 3 ) ^ 2 ]
cos ( theta ) = + OR - ( 4 / 3 ) / sqrt ( 1 + 16 / 9 )
cos ( theta ) = + OR - ( 4 / 3 ) / sqrt ( 9 / 9 + 16 / 9 )
cos ( theta ) = + OR - ( 4 / 3 ) / sqrt ( 25 / 9 )
cos ( theta ) = + OR - ( 4 / 3 ) / ( 5 / 3 )
cos ( theta ) = + OR - ( 4 * 3 ) / ( 3 * 5 )
cos ( theta ) = + OR - 4 / 5
cot ( theta ) = cos ( theta ) / sin ( theta )
In quadran I cosine and sine are positive so quotient of cosine and sine will be positive.
In quadran III cosine and sine are negative so quotient of cosine and sine will be positive.
Now you have 2 sets of solutions :
1. In your equation put :
sin ( theta ) = 3 / 5
cos ( theta ) = 4 / 5
2. In your equation put :
sin ( theta ) = - 3 / 5
cos ( theta ) = - 4 / 5
If your e3xpression mean :
5 cos ( theta ) - 2 sin ( theta ) / [ 5 cos ( theta ) ] + 3 sin ( theta )
Solutions will bee :
- 61 / 10
and
11 / 2
To find the value of the expression (5cos theta - 2sin theta) / (5cos theta + 3sin theta) when 3cot theta = 4, we need to solve for theta.
First, let's simplify the equation 3cot theta = 4:
Divide both sides of the equation by 3:
cot theta = 4/3
Since cot theta is the reciprocal of tan theta, we can rewrite the equation as:
tan theta = 3/4
Now, we can find the value of theta by taking the inverse tangent (or arctan) of both sides:
theta = arctan(3/4)
Using a scientific calculator or trigonometric table, you can find the value of arctan(3/4). Let's assume it is equal to x degrees.
Now, substituting the value of theta in the expression (5cos theta - 2sin theta) / (5cos theta + 3sin theta):
(5cos x - 2sin x) / (5cos x + 3sin x)
The value of this expression will depend on the specific value of x that we obtained from arctan(3/4).
So, simplify the expression and substitute the value of x obtained from arctan(3/4) to find the final answer to the given expression.