write each expression as the sine, cosine or tangent of a double angle. then find the exact value of the expression.
a. 2sin 22.5 cos 22.5
b. cos^2 105- sin^2 105
a, use
sin 45 = cos 45 = 1/sqrt 2
b, use
105 = (1/2) 210 which is 180 + 30
so use
sin 210 = -1/2
cos 210 = -sqrt 3/2
a. We can write the expression 2sin 22.5 cos 22.5 as the sine of a double angle. The double angle identity for sine is:
sin(2θ) = 2sin(θ)cos(θ)
In this case, θ is equal to 22.5 degrees. So, we have:
2sin(22.5)cos(22.5) = sin(2 * 22.5)
Now, let's evaluate sin(45):
sin(45) = √2 / 2
Therefore:
2sin(22.5)cos(22.5) = sin(2 * 22.5) = sin(45) = √2 / 2
b. We can write the expression cos^2(105) - sin^2(105) as the cosine of a double angle. The double angle identity for cosine is:
cos(2θ) = cos^2(θ) - sin^2(θ)
In this case, θ is equal to 105 degrees. So, we have:
cos(2 * 105) = cos^2(105) - sin^2(105)
Now, let's evaluate cos(210):
cos(210) = -√3 / 2
Therefore:
cos^2(105) - sin^2(105) = cos(2 * 105) = cos(210) = -√3 / 2
a. To write the expression as the sine or cosine of a double angle, we can use the double angle formulas:
1. For sine: sin(2θ) = 2sin(θ)cos(θ)
2. For cosine: cos(2θ) = cos^2(θ) - sin^2(θ)
So for the given expression, 2sin 22.5 cos 22.5, we can use the double angle formula for sine:
2sin(22.5)cos(22.5) = sin(2(22.5))
Now, we can find the exact value of sin(2(22.5)) using trigonometric identities and the angles we know:
We can write 22.5 as the sum of 45/2 degrees and use the half-angle formula for sine:
sin(2(22.5)) = sin(2*(45/2))
= sin(45)
= 1/√2
Therefore, the exact value of the expression 2sin 22.5 cos 22.5 is 1/√2.
b. To write the expression cos^2 105 - sin^2 105 as the cosine of a double angle, we can use the double angle formula for cosine:
cos(2θ) = cos^2(θ) - sin^2(θ)
Now let's substitute θ with 105 in the formula:
cos(2(105)) = cos^2(105) - sin^2(105)
To find the exact value, we need to find the values of cos^2(105) and sin^2(105).
Using a trigonometric identity, we know that cos^2(θ) + sin^2(θ) = 1. Therefore, we can rewrite the expression as:
cos^2(105) - sin^2(105) = 1 - sin^2(105) - sin^2(105)
Since sin^2(θ) + cos^2(θ) = 1, we can substitute sin^2(105) with 1 - cos^2(105):
1 - sin^2(105) - sin^2(105) = 1 - (1 - cos^2(105)) - (1 - cos^2(105))
Now, we simplify:
1 - (1 - cos^2(105)) - (1 - cos^2(105)) = 1 - 1 + cos^2(105) - 1 + cos^2(105)
The 1s cancel out:
1 + cos^2(105) - 1 + cos^2(105) = 2cos^2(105)
Therefore, the exact value of the expression cos^2 105 - sin^2 105 is 2cos^2(105).