Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
2tan 2pi/3 / 1-tan^2 2pu/3
that would be tan 4π/3 = √3
To express the given expression in terms of a double angle, we can rewrite it using trigonometric identities.
Starting with the numerator, 2tan(2π/3), we know the tangent double angle identity: tan(2θ) = 2tan(θ) / (1 - tan^2(θ)).
So, the numerator can be expressed as: 2tan(2π/3) = tan(2(π/3)) = tan(2π/3).
Moving on to the denominator, (1 - tan^2(2π/3)), we can further simplify it using the Pythagorean identity: tan^2(θ) + 1 = sec^2(θ).
Therefore, 1 - tan^2(θ) = 1 - sec^2(θ) = -sec^2(θ).
Now, we can rewrite the original expression as the sine, cosine, or tangent of a double angle:
2tan(2π/3) / (1 - tan^2(2π/3)) = tan(2π/3) / (-sec^2(2π/3)).
To find the exact value, we need to substitute the values of the trigonometric functions at 2π/3. Let's begin by finding the value of tan(2π/3):
In a unit circle, at angle 2π/3, the y-coordinate is √3/2, and the x-coordinate is -1/2. Therefore, tan(2π/3) = (y-coordinate) / (x-coordinate) = (√3/2) / (-1/2) = -√3.
Next, we find the value of sec(2π/3):
Using the Pythagorean identity, sec^2(θ) = tan^2(θ) + 1.
For 2π/3, sec^2(2π/3) = tan^2(2π/3) + 1 = (-√3)^2 + 1 = 3 + 1 = 4.
Therefore, sec(2π/3) = √4 = 2.
Now, we have the value of tan(2π/3) and sec(2π/3), so we can substitute them into the expression:
tan(2π/3) / (-sec^2(2π/3)) = -√3 / (-2^2) = -√3 / 4.
Hence, the exact value of the given expression is -√3 / 4.