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, we need to use trigonometric identities. Let's solve each problem step by step:
1. cos^2(pi/8) - 1/2:
To express this as a single trigonometric function value, we can use the double-angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.
In this case, θ = pi/8. Using the double-angle identity, we can rewrite cos^2(pi/8) as (1 + cos(2(pi/8))) / 2.
Therefore, the expression becomes: (1 + cos(2(pi/8))) / 2 - 1/2.
2. tan(34 degrees) / 2(1 - tan^2(34 degrees)):
To express this as a single trigonometric function value, we can use the identity for tangent squared: tan^2(θ) = sec^2(θ) - 1.
In this case, θ = 34 degrees. Using the tangent squared identity, we can rewrite tan^2(34 degrees) as sec^2(34 degrees) - 1.
Therefore, the expression becomes: tan(34 degrees) / 2(1 - (sec^2(34 degrees) - 1)).
Next, we use the identity for secant squared: sec^2(θ) = 1 + tan^2(θ). So, sec^2(34 degrees) = 1 + tan^2(34 degrees).
Substituting this into our expression: tan(34 degrees) / 2(1 - ((1 + tan^2(34 degrees)) - 1)).
Now, we simplify further: tan(34 degrees) / 2(1 - tan^2(34 degrees) + 1 - 1).
Finally, we get: tan(34 degrees) / (2 - 2tan^2(34 degrees)).
3. (1/8)sin(29.5 degrees)cos(29.5 degrees):
To express this as a single trigonometric function value, we can use the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ).
In this case, θ = 29.5 degrees. Using the double-angle identity, we can rewrite sin(29.5 degrees)cos(29.5 degrees) as (1/2)sin(59 degrees).
Finally, the expression becomes: (1/8)(1/2)sin(59 degrees) = (1/16)sin(59 degrees).
Note that we simplify (1/8)(1/2) as (1/16).
So, the expressions in terms of single trigonometric function values or single numbers using identities are:
1. (1 + cos(2(pi/8)))/2 - 1/2
2. tan(34 degrees) / (2 - 2tan^2(34 degrees))
3. (1/16)sin(59 degrees)