a. Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine:
(cos^4)x (or cos*x to the 4th power)
b.Use an appropriate Half- Angle Formula to Find the Exact value of the expression:
1. tan 15(Degrees)
2. cos 3pi/8(Radians)
cos^2 x = (1+cos2x)/2, so
cos^4 x = (1 + 2cos2x + cos^2 2x)/4
= (1 + 2cos2x + (1+cos4x)/2))/4
= (3 + 4cos2x + cos4x)/8
what do you get for the others?
1.square root 2 - 1 or 2^1/2 -1
2.I got -1/2 2^1/2-2^1/4 or -1/2 *square root 2 minus square root 2 (the second 2 has two square roots over it)
a. To rewrite the expression (cos^4)x in terms of the first power of cosine, we can use the lowering powers formula for cosine:
cos^2(x) = (1 + cos(2x))/2
Using this formula, we can express cos^4(x) as:
cos^4(x) = (cos^2(x))^2 = ((1 + cos(2x))/2)^2
b.
1. To find the exact value of tan 15 degrees, we can use the half-angle formula for tangent:
tan(x/2) = sqrt( (1 - cos(x)) / (1 + cos(x)) )
Using this formula, we can find the exact value of tan 15 degrees:
tan 15° = tan(30°/2) = sqrt( (1 - cos(30°)) / (1 + cos(30°)) )
Now, we can use the values from the unit circle to find cos(30°). From the unit circle, we know that cos(30°) = sqrt(3)/2.
Plugging in this value, we get:
tan 15° = sqrt( (1 - sqrt(3)/2) / (1 + sqrt(3)/2) )
Simplifying this expression further will give us the exact value of tan 15 degrees.
2. To find the exact value of cos(3π/8) radians, we can use the half-angle formula for cosine:
cos(x/2) = sqrt( (1 + cos(x)) / 2 )
Using this formula, we can find the exact value of cos(3π/8) radians:
cos(3π/8) = cos((3π/4)/2) = sqrt( (1 + cos(3π/4)) / 2 )
Now, we can use the values from the unit circle to find cos(3π/4). From the unit circle, we know that cos(3π/4) = -sqrt(2)/2.
Plugging in this value, we get:
cos(3π/8) = sqrt( (1 - sqrt(2)/2) / 2 )
Simplifying this expression further will give us the exact value of cos(3π/8) radians.