5.07 Power-Reducing and Half-Angle Formulas

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Power-Reducing and Half-Angle Formulas Practice

Practice 1

Use a half-angle formula to find the exact value of pi over 12

$the tangent of x over 2 equals the fraction with numerator 1 minus blank square x and denominator blank x$
Select the correct half-angle formula

Answer:
$the tangent of x over 2 equals the fraction with numerator 1 minus cosine x and denominator sine x$
Let = . Then x =
Answer:

Let = . Then x =

$the tangent of open paren the fraction with numerator pi and denominator blank square close paren equals the fraction with numerator 1 minus the cosine of pi over blank and denominator the sine of pi over blank$

Substitute in x

Answer:

$the tangent of open paren pi over 12 close paren equals the fraction with numerator 1 minus the cosine of the fraction with numerator pi and denominator pi over 6 and denominator the sine of pi over 6$

Use a half-angle formula to find the exact value of tangent pi over 12

$3 lines Line 1: the tangent of x over 2 equals the fraction with numerator 1 minus cosine x and denominator sine x Line 2: Let x over 2 equals pi over 12 then x equals pi over 6 Line 3: the tangent of open paren pi over 12 close paren equals the fraction with numerator 1 minus the cosine of pi over 6 and denominator the sine of pi over 6$

Select the correct half-angle formula

Substitute in x

$the tangent of open paren pi over 12 close paren equals the fraction with numerator 1 minus the fraction with numerator the square root of blank and denominator blank and denominator the fraction with numerator blank and denominator blank$

Substitute the value of sin pi over 6 and cos

Answer:

$the tangent of open paren pi over 12 close paren equals the fraction with numerator 1 minus the fraction with numerator the square root of 3 and denominator 2 and denominator the fraction with numerator 1 and denominator 2$

the tangent of open paren pi over 12 close paren equals blank square minus the square root of blank square

Multiply by common denominator

Answer:

open paren 1 blank square cosine 2 x over 2 close paren times open paren 1 blank square cosine 2 x over 2 close paren

Practice #2, Part 1

Rewrite the expression in terms of the first power of the cosine.

cos⁴x

____ (Fill in the blank)²x ____ (Fill in the blank)²x
Since we don’t have any formulas for the fourth power of cosine, start by factoring

Answer:

cos²x cos²x

Now, we can substitute power-reducing formulas

Answer:

$open paren 1 plus cosine 2 x over 2 close paren times open paren the fraction with numerator 1 plus cosine 2 x and denominator 2 close paren$

$the fraction with numerator blank plus cosine 2 x plus cosine 2 x plus blank squared 2 x and denominator 4$

Multiply

Answer:

$the fraction with numerator 1 plus cosine 2 x plus cosine 2 x plus cosine squared times 2 x and denominator 4$

$the fraction with numerator 1 plus 2 times cosine 2 x plus cosine squared 2 x and denominator 4$

Add like terms

Practice #2, Part 2

Rewrite the expression in terms of the first power of the cosine.

cos⁴x

$4 lines Line 1: cosine squared x cosine squared x Line 2: open paren the fraction with numerator 1 plus cosine 2 x and denominator 2 close paren times open paren the fraction with numerator 1 plus cosine 2 x and denominator 2 close paren Line 3: the fraction with numerator 1 plus cosine 2 x plus cosine 2 x plus cosine squared 2 x and denominator 4 Line 4: the fraction with numerator 1 plus 2 cosine 2 x plus cosine squared 2 x and denominator 4$
We’ve gotten another cos² x term. We need to substitute the power reducing formula again.

$the fraction with numerator 1 plus 2 cosine 2 x plus the fraction with numerator 1 blank square cosine 2 open paren blank square close paren and denominator 2 and denominator 4$

Answer:

$the fraction with numerator 1 plus 2 cosine 2 x plus the fraction with numerator 1 plus cosine 2 open paren 2 x close paren and denominator 2 and denominator 4$

Simplify cos 2(2x)

Answer:

$the fraction with numerator 1 plus 2 cosine 2 x plus the fraction with numerator 1 plus cosine 4 x and denominator 2 and denominator 4$

Practice #2, Part 3

Rewrite the expression in terms of the first power of the cosine.

cos⁴x

$3 lines Line 1: cosine squared x cosine squared x Line 2: the fraction with numerator 1 plus 2 cosine 2 x plus cosine squared 2 x and denominator 4 Line 3: the fraction with numerator 1 plus 2 cosine 2 x plus the fraction with numerator 1 plus the cosine of 2 times 2 x and denominator 2 and denominator 4$
$the fraction with numerator open paren 1 plus 2 cosine 2 x plus the fraction with numerator 1 plus cosine 4 x and denominator 2 close paren and denominator 4 times two halves$

Multiply by the common denominator.

$the fraction with numerator blank square plus blank square cosine 2 x plus blank square plus cosine 4 x and denominator blank square$

Answer:

$the fraction with numerator 2 plus 4 cosine 2 x plus 1 plus cosine 4 x and denominator 8$

Practice #2, Part 4

Rewrite the expression in terms of the first power of the cosine.

cos⁴x

$2 lines Line 1: the fraction with numerator open paren 1 plus 2 cosine 2 x plus the fraction with numerator 1 plus cosine 4 x and denominator 2 close paren and denominator 4 times two halves Line 2: the fraction with numerator 2 plus 4 cosine 2 x plus 1 plus cosine 4 x and denominator 8$
$the fraction with numerator blank plus blank cosine blank plus cosine blank and denominator 8$

Simplify

Answer:

$the fraction with numerator 3 plus 4 cosine 2 x plus cosine 4 x and denominator 8$