if tan(x)=9/4, find cos(x) if pi<x<3pi/2

pi < x < 3 pi / 2

180 ° < x < 270 °

That is Quadran III

In Quadran III cosine are negative.

cos ( x ) = + OR - 1 /sqrt [ 1 + tan( x ) ^ 2 ]

In this case :

cos ( x ) = - 1 / sqrt [ 1 + tan( x ) ^ 2 ]

cos ( x ) = - 1 / sqrt [ 1 + ( 9 / 4 ) ^ 2 ]

cos ( x ) = - 1 / sqrt ( 1 + 81 / 16 )

cos ( x ) = - 1 / sqrt ( 16 / 16 + 81 / 16 )

cos ( x ) = - 1 / sqrt ( 97 / 16 )

cos ( x ) = - 1 / [ sqrt ( 97 ) / 4 ]

cos ( x ) = - 4 / sqrt ( 97 )

Thanks(:

To find cos(x) if tan(x) = 9/4 and pi < x < 3pi/2, we can use the relationship between tangent and cosine.

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle. However, since we are dealing with angles in quadrant II (pi < x < 3pi/2), where the tangent is positive and the cosine is negative, we need to use the negative value of the tangent ratio.

Given that tan(x) = 9/4, we have the following information:
Opposite side = 9
Adjacent side = 4

Now, we can use the Pythagorean theorem to find the hypotenuse of the right triangle. The Pythagorean theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, the hypotenuse is represented by the square root of the sum of the squares of the opposite and adjacent sides.

Using the Pythagorean theorem:
c^2 = 9^2 + 4^2
c^2 = 81 + 16
c^2 = 97

Taking the square root of both sides:
c = sqrt(97)

Now that we have the lengths of all three sides of the triangle, we can find the cosine of x using the adjacent side and the hypotenuse:

cos(x) = adjacent side / hypotenuse
cos(x) = 4 / sqrt(97)

Therefore, cos(x) = 4 / sqrt(97), where pi < x < 3pi/2.