Question

determine the value of cos2x when tanx=3/4 and pi<x<3pi/2

Answer 1

We know that tan(x) = 3/4. Since x is in the second quadrant (pi < x < 3pi/2), we know that x is negative and the x-coordinate is negative.

Using the Pythagorean identity, we have:

tan^2(x) + 1 = sec^2(x)

(3/4)^2 + 1 = sec^2(x)

9/16 + 1 = sec^2(x)

25/16 = sec^2(x)

Taking the square root of both sides, we have:

√(25/16) = √(sec^2(x))

5/4 = sec(x)

Since the x-coordinate is negative in the second quadrant, the secant function is negative. Therefore, sec(x) = -5/4.

To find cos(2x), we can use the double-angle identity for cosine:

cos(2x) = 1 - 2sin^2(x)

Since tan(x) = 3/4, we can find sin(x) using the Pythagorean identity:

sin(x) = √(1 - cos^2(x))

sin(x) = √(1 - (3/5)^2)

sin(x) = √(1 - 9/25)

sin(x) = √(16/25)

sin(x) = 4/5

Now we can substitute the values of sin(x) and cos(x) into the formula for cos(2x):

cos(2x) = 1 - 2(4/5)^2

cos(2x) = 1 - 2(16/25)

cos(2x) = 1 - 32/25

cos(2x) = (25/25) - (32/25)

cos(2x) = -7/25

Therefore, cos(2x) = -7/25.