How do you express these problems as single trigonometric function values or as single numbers, using identities?
1. cos^2pi/8-1/2
2. tan34degrees/2(1-tan^2 34degrees)
3. 1/8sin29.5degreescos29.5degrees
To express these problems as single trigonometric function values or as single numbers using identities, we need to apply trigonometric identities to simplify the expressions.
1. cos^2(pi/8) - 1/2:
We can use the double angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.
In this case, θ = π/8.
By rearranging the formula, we have cos^2(θ) = (1 + cos(2θ))/2.
Now substitute π/8 as θ:
cos^2(π/8) - 1/2 = (1 + cos(2π/8))/2 - 1/2.
Simplifying, we find:
cos^2(π/8) - 1/2 = cos(π/4)/2 - 1/2 = (sqrt(2)/2)/2 - 1/2 = sqrt(2)/4 - 1/2.
2. tan(34°)/2(1 - tan^2(34°)):
To simplify this expression, we can use the tangent double-angle identity: tan(2θ) = (2tan(θ))/(1 - tan^2(θ)).
Let θ = 34°.
Now, we can rewrite the expression as:
tan(34°)/2(1 - tan^2(34°)) = tan(68°)/(1 - tan^2(34°)).
Using the double-angle identity for tangent, we have:
tan(68°)/(1 - tan^2(34°)) = (2tan(34°))/(1 - (tan(34°))^2).
Now substitute tan(34°) = x to simplify further:
tan(68°)/(1 - tan^2(34°)) = (2x)/(1 - x^2).
3. (1/8)sin(29.5°)cos(29.5°):
We can use the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ).
Letting θ = 29.5°, we have:
(1/8)sin(29.5°)cos(29.5°) = (1/8)(2sin(29.5°)cos(29.5°)/2).
Using the double-angle identity, we simplify it to:
(1/8)(2sin(29.5°)cos(29.5°)/2) = (1/8)sin(59°)/2 = (1/16)sin(59°).