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)
Create a program that uses the same formula above to output
the result; this time, however, prompt the user for the values
a, b, x, and y. Use appropriate variable names and naming
conventions.
a. To rewrite the expression (cos^4)x in terms of the first power of cosine, we can use the formula for lowering powers:
cos^4(x) = (cos^2(x))^2
Now, let's further simplify this. We can use the formula for the square of cosine:
cos^2(x) = (1 + cos(2x))/2
Substituting this back into the previous equation, we have:
(cos^4)x = [(1 + cos(2x))/2]^2
This is the expression rewritten in terms of the first power of cosine.
b. To find the exact values of the expressions tan 15 degrees and cos 3pi/8 radians using a Half-Angle Formula, we first need to identify the appropriate Half-Angle Formula for each expression.
1. tan 15 degrees:
We can use the Half-Angle Formula for tangent, which is:
tan(x/2) = (1 - cos(x)) / sin(x)
Here, x is the given angle (in this case, 15 degrees). So, substituting x = 15 degrees into the formula, we have:
tan(15/2) = (1 - cos(15)) / sin(15)
Now, we need to find the values of cos(15) and sin(15). To do this, we can use a calculator or look up the values in a trigonometric table. Once we have these values, we can substitute them into the formula and simplify to find the exact value of tan 15 degrees.
2. cos 3pi/8 radians:
We can use the Half-Angle Formula for cosine, which is:
cos(x/2) = ±sqrt((1 + cos(x))/2)
Here, x is the given angle (in this case, 3pi/8 radians). So, substituting x = 3pi/8 radians into the formula, we have:
cos((3pi/8)/2) = ±sqrt((1 + cos(3pi/8))/2)
Now, we need to find the value of cos(3pi/8). We can use a calculator or look up the value in a trigonometric table. Once we have this value, we can substitute it into the formula and simplify to find the exact value of cos 3pi/8 radians.