Given that pi < x < 2pi and tanx=3/4, determine the exact value of cos2x. Show all work including a diagram, special triangles, CAST rules and related acute angles for full marks.
To solve this problem, we need to use the information provided to find the exact value of cos(2x). Let's go step by step.
Step 1: Draw a diagram
Draw a diagram of a unit circle to visualize the situation. Place the angles x and 2x in the quadrant where pi and 2pi lie, ensuring that both angles satisfy the given condition pi < x < 2pi.
Step 2: Determine the related acute angle
Since we are given that tan(x) = 3/4, we can determine the related acute angle by using the inverse tangent function. Therefore, we have:
tan(arc tan(3/4)) = tan(θ), where θ is the acute angle opposite the sides of length 3 and 4 in a right triangle.
Let's solve this equation:
θ = arc tan(3/4).
Using a calculator, we find θ ≈ 36.87 degrees.
Step 3: Determine the quadrant
Since the tangent is positive (3/4), the angle θ is in the first quadrant.
Step 4: Use the CAST rule
We need to determine whether x and 2x lie in the same or different quadrants. Since pi < x < 2pi, we know that x lies in the second quadrant. When we double that angle, 2x, it will lie in the fourth quadrant.
Step 5: Determine the value of cos(x)
Since x lies in the second quadrant and its tangent is 3/4, we can determine the missing side in the right triangle formed. Since tan(x) = opposite/adjacent, we know that the opposite side has a length of 3 and the adjacent side has a length of 4. Then, using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse = sqrt(opposite^2 + adjacent^2)
= sqrt(3^2 + 4^2)
= 5.
Therefore, in the second quadrant, cos(x) = adjacent/hypotenuse = 4/5.
Step 6: Determine the value of cos(2x)
Since we know the value of cos(x), we can use the double angle formula to find cos(2x):
cos(2x) = cos^2(x) - sin^2(x).
sin^2(x) can be determined using sin^2(x) = 1 - cos^2(x) in the second quadrant:
sin^2(x) = 1 - (4/5)^2
= 1 - 16/25
= 9/25.
Therefore, cos(2x) = cos^2(x) - sin^2(x)
= (4/5)^2 - 9/25
= 16/25 - 9/25
= 7/25.
So, the exact value of cos(2x) for the given conditions is 7/25.