use an appropriate half-angle formula to find the exact value ofthe expression.
tan22.5
write the product as a sum
sinxsin5x
write the sum as a product
sin2x-sin7x
sinxsin5x
sinucosv= 1/2[cos(u-v)-cos(u+v)]
= 1/2[cos(x-5x)-cos(x+5x)]
= 1/2[cos(-4x)-cos(6x)]
= 1/2cos4x-1/2cos6x
To find the exact value of the expression tan(22.5), we can use the half-angle formula for tangent.
The half-angle formula for tangent is given by:
tan(x/2) = sqrt((1 - cos(x)) / (1 + cos(x)))
To find the value of tan(22.5), we can substitute x = 45 degrees into the formula:
tan(22.5) = tan(45/2)
= sqrt((1 - cos(45)) / (1 + cos(45)))
cos(45) can be obtained from the special 45-45-90 degree triangle, where the adjacent and opposite sides are equal in length. Therefore, cos(45) = 1/√2.
Substituting this value into the formula:
tan(22.5) = sqrt((1 - 1/√2) / (1 + 1/√2))
To rationalize the denominator, we multiply both the numerator and the denominator by √2:
tan(22.5) = sqrt((√2 - 1) / (√2 + 1)) * √2/√2
= sqrt(2 - √2) / sqrt(2 + √2)
= (sqrt(2 - √2) * sqrt(2 - √2)) / (sqrt(2 + √2) * sqrt(2 - √2))
= (2 - √2) / (2 + √2)
Therefore, the exact value of tan(22.5) is (2 - √2) / (2 + √2).
Moving on to the next question,
To write the product sin(x) * sin(5x) as a sum, we can use the identity:
sin(a) * sin(b) = (cos(a - b) - cos(a + b)) / 2
Applying this identity, with a = x and b = 5x, we have:
sin(x) * sin(5x) = (cos(x - 5x) - cos(x + 5x)) / 2
= (cos(-4x) - cos(6x)) / 2
= (cos(4x) - cos(6x)) / 2
Therefore, sin(x) * sin(5x) can be written as (cos(4x) - cos(6x)) / 2.
Finally, for the expression sin(2x) - sin(7x), we can write it as a product using the identity:
sin(a) - sin(b) = 2 * cos((a + b)/2) * sin((a - b)/2)
Applying this identity, with a = 2x and b = 7x, we have:
sin(2x) - sin(7x) = 2 * cos((2x + 7x)/2) * sin((2x - 7x)/2)
= 2 * cos(9x/2) * sin(-5x/2)
= -2 * cos(9x/2) * sin(5x/2)
Therefore, sin(2x) - sin(7x) can be written as -2 * cos(9x/2) * sin(5x/2).