1. Find the exact value of 2tan pi/12 / 1-tan^2 pi/12
root 3/3?
2. Given tanθ = -1/3 and with θ in quadrant IV, find the exact value of cos2(θ)
4/5
3. Exact value of cos^2 67.5 - sin^2 67.5
root3 /2
4. The expression cos 3θ can be rewritten as 1 - 4cosθ + 4cos^3θ
False
5. The expression sin 3θ can be rewritten as 3sinθ - 4sin^3 θ
Not sure
#1. ok
#2. find sinθ and cosθ and let 'er rip
#3. you have cos 135° = -1/√2
cos2θ = cos^2 θ - sin^2 θ
#4. Obviously false, since if θ=90 we have
cos3θ = 0
1-4cosθ+4cos^3θ = 1-0+0 = 1
#5. True
sin3θ = sin2θcosθ + cos2θsinθ
= 2sinθcos^2θ + sinθ - 2sin^3θ
= 2sinθ - 2sin^3θ + sinθ - 2sin^3θ
= 3sinθ - 4sin^3θ
Thank you for your constant help! I appreciate it :)
1. To find the exact value of 2tan(pi/12) / (1-tan^2(pi/12)), we can start by using the tangent angle addition formula. The formula states that tan(A + B) = (tan A + tan B) / (1 - tan A tan B).
In this case, we have A = B = pi/12. Plugging it into the formula, we get:
tan(2(pi/12)) = (tan(pi/12) + tan(pi/12)) / (1 - tan(pi/12) * tan(pi/12))
Since tan(pi/12) is not a common angle, we can use the exact values from the unit circle. We know that sin(pi/6) = sqrt(3)/2 and cos(pi/6) = 1/2.
Using the identity tan(theta) = sin(theta) / cos(theta), we can find tan(pi/6) = (sqrt(3)/2) / (1/2) = sqrt(3).
Plugging it back into the equation, we get:
tan(2(pi/12)) = (sqrt(3) + sqrt(3)) / (1 - sqrt(3) * sqrt(3))
tan(2(pi/12)) = 2sqrt(3) / (1 - 3)
tan(2(pi/12)) = 2sqrt(3) / -2
Canceling out the 2, we get:
tan(2(pi/12)) = -sqrt(3)
Therefore, the exact value of 2tan(pi/12) / (1-tan^2(pi/12)) is -sqrt(3).
2. Since we are given that tan(theta) = -1/3 and theta is in quadrant IV, we can determine that sin(theta) is negative (since it is negative in quadrant IV) and cos(theta) is positive (since it is positive in quadrant IV).
Using the identity tan(theta) = sin(theta) / cos(theta), we can find sin(theta) = -1 and cos(theta) = 3.
To find cos^2(theta), we square the value of cos(theta):
cos^2(theta) = (cos(theta))^2 = (3/5)^2 = 9/25
To find cos(2(theta)), we can use the double-angle formula for cosine, which states that cos(2(theta)) = cos^2(theta) - sin^2(theta).
Plugging in the values we found earlier, we get:
cos(2(theta)) = (9/25) - (-1)^2 = 9/25 - 1 = 9/25 - 25/25 = -16/25
Therefore, the exact value of cos2(theta) is -16/25.
3. To find the exact value of cos^2(67.5) - sin^2(67.5), we can use the double-angle formula for cosine, which states that cos(2(theta)) = cos^2(theta) - sin^2(theta).
Plugging in the given value of 67.5 degrees, we get:
cos^2(67.5) - sin^2(67.5) = cos(2 * 67.5)
Using the double-angle identity for cosine, we have:
cos(2 * 67.5) = cos(135) = cos(180 - 45)
Since we know that cos(180 - theta) = -cos(theta), we can rewrite it as:
cos(135) = -cos(45)
Using the exact values of cos(45) = sqrt(2)/2, we get:
cos(135) = -sqrt(2)/2
Therefore, the exact value of cos^2(67.5) - sin^2(67.5) is -sqrt(2)/2.
4. The given statement that the expression cos(3θ) can be rewritten as 1 - 4cos(θ) + 4cos^3(θ) is false. The correct expansion of cos(3θ) is:
cos(3θ) = cos(θ + θ + θ)
Using the triple-angle formula for cosine, we have:
cos(3θ) = cos^3(θ) - 3cos(θ)sin^2(θ)
Therefore, the given statement is false.
5. The expression sin(3θ) can be rewritten using the triple-angle formula for sine as:
sin(3θ) = 3sin(θ) - 4sin^3(θ)
Therefore, the statement that the expression sin(3θ) can be rewritten as 3sin(θ) - 4sin^3(θ) is true.