To determine the exact value of cos(2x), we can use the identity:
cos(2x) = cos^2(x) - sin^2(x)
First, we need to find the value of sin(x) using the given equation tan(x) = 3/4.
Using the definition of tan (tan(x) = sin(x) / cos(x)), we have:
3/4 = sin(x) / cos(x)
Cross-multiplying gives us:
4 * sin(x) = 3 * cos(x)
Dividing both sides by cos(x), we get:
sin(x) / cos(x) = 3/4
Using the Pythagorean identity (sin^2(x) + cos^2(x) = 1), we know that:
sin^2(x) = 1 - cos^2(x)
Substituting the value of sin(x) from the previous equation, we can solve for cos(x):
(3/4)^2 = 1 - cos^2(x)
9/16 = 1 - cos^2(x)
Rearranging the equation, we get:
cos^2(x) = 1 - 9/16
cos^2(x) = 16/16 - 9/16
cos^2(x) = 7/16
Taking the square root of both sides, we obtain:
cos(x) = ±√(7/16)
Since π < x < 2π, x lies in the second quadrant where cos(x) is negative. Therefore, we take the negative root:
cos(x) = -√(7/16)
We can relate the given interval to the unit circle as follows:
- π < x < 2π means 180° < x < 360°
Using CAST (where C stands for cos, A for all trigonometric functions, S for sin, T for tan):
- In the second quadrant, sin and tan are positive, cos is negative.
- In the third and fourth quadrants, all trigonometric functions are negative.
We can draw a diagram of the unit circle in the second quadrant:
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In the second quadrant, the reference angle is α = 180° - x.
Using the special triangles (45-45-90 and 30-60-90 triangles), we can find the value of cos(x) and sin(x). For reference, the 45-45-90 triangle has sides in the ratio 1:1:√2, and the 30-60-90 triangle has sides in the ratio 1:√3:2.
Since sin(x) = 3/4, we can find cos(x):
sin(x) = 3/4
Using the Pythagorean identity (sin^2(x) + cos^2(x) = 1), we have:
(3/4)^2 + cos^2(x) = 1
9/16 + cos^2(x) = 1
cos^2(x) = 1 - 9/16
cos^2(x) = 16/16 - 9/16
cos^2(x) = 7/16
Taking the square root of both sides, we get:
cos(x) = ±√(7/16)
Since x lies in the second quadrant, where cos(x) is negative:
cos(x) = -√(7/16)
Now, to find cos(2x), we can use the double angle identity:
cos(2x) = cos^2(x) - sin^2(x)
Substituting the values we found:
cos(2x) = (-√(7/16))^2 - (3/4)^2
cos(2x) = 7/16 - 9/16
cos(2x) = -2/16
Simplifying, we get:
cos(2x) = -1/8
Therefore, the exact value of cos(2x) is -1/8.