5.07 Power-Reducing and Half-Angle Formulas

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Power-Reducing Formulas

Let’s look at the cos 2x formulas and use them to derive the power-reducing formulas:

cos 2x = 1 − 2 sin²x

Subtract 1 from both sides

cos 2x = 1 − 2 sin²x

Divide by −2

$the fraction with numerator cosine 2 x minus 1 and denominator negative 2 equals sine squared x$

Rearrange

$sine squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 2$

Power-Reducing Formulas

Let’s look at the cos 2x formulas and use them to derive the power-reducing formulas:

cos 2x = 2 cos²x − 1

Add 1 to both sides

cos 2x + 1 = 2 cos²x

Divide by 2

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

Rearrange

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

Power-Reducing Formulas

In order to find the tan²x let’s use the quotient identity tangent x equals sine x over cosine x

Substitute the formulas we just found

$tangent squared x equals the fraction with numerator the fraction with numerator 1 minus cosine 2 x and denominator 2 and denominator the fraction with numerator cosine 2 x plus 1 and denominator 2$

Simplify

$tangent squared x equals the fraction with numerator 1 minus cosine 2 x and denominator cosine 2 x plus 1$

The Power-Reducing Formulas

$sine squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 2$
$cosine squared x equals the fraction with numerator 1 plus cosine 2 x and denominator 2$
$tangent squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 1 plus cosine 2 x$

Half-Angle Formulas

The half-angle formulas are derived from the power-reducing formulas.

$sine squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 2$

Replace x with x over 2

$the sine squared of x over 2 equals the fraction with numerator 1 minus cosine x and denominator 2$

Take the square root of both sides

$the sine of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine x and denominator 2$

Notice that 2x becomes x when x is replaced with x over 2

Half-Angle Formulas

The half-angle formulas are derived from the power-reducing formulas.

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

Replace x with x over 2

$the cosine squared of x over 2 equals the fraction with numerator 1 plus cosine x and denominator 2$

Take the square root of both sides

$the cosine squared of x over 2 equals the fraction with numerator 1 plus cosine x and denominator 2$

Notice that 2x becomes x when x is replaced with x over 2

Half-Angle Formulas

The half-angle formulas are derived from the power-reducing formulas.

$tangent squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 1 plus cosine 2 x$

Replace x with x over 2

$the tangent squared of x over 2 equals the fraction with numerator 1 minus cosine x and denominator 1 plus cosine x$

Take the square root of both sides

$the tangent of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine x and denominator 1 plus cosine x$

We now need to simplify the radical.

Simplifying the Tangent Half-Angle Formula

$the tangent of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine x and denominator 1 plus cosine x$

To simplify, multiply either by the conjugate of the denominator, or by the conjugate of the numerator.

Conjugate of the denominator		Conjugate of the numerator
$the tangent of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine x and denominator 1 plus cosine x times open paren the fraction with numerator 1 minus cosine x and denominator 1 minus cosine x close paren$		$the tangent of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine x and denominator 1 plus cosine x times open paren the fraction with numerator 1 plus cosine x and denominator 1 plus cosine x close paren$
$3 lines Line 1: the tangent of x over 2 equals plus or minus the square root of the fraction with numerator open paren 1 minus cosine x close paren squared and denominator open paren 1 minus cosine x close paren squared Line 2: the tangent of x over 2 equals plus or minus the square root of the fraction with numerator open paren 1 minus cosine x close paren squared and denominator sine squared x Line 3: the tangent of x over 2 equals plus or minus the fraction with numerator 1 minus cosine x and denominator sine x$	Pythagorean identity sin²x + cos²x = 1 Find the square root	$3 lines Line 1: the tangent of x over 2 equals plus or minus the square root of the fraction with numerator 1 minus cosine squared x and denominator open paren 1 plus cosine x close paren squared Line 2: the tangent of x over 2 equals plus or minus the square root of the fraction with numerator sine squared x and denominator open paren 1 plus cosine x close paren squared Line 3: the tangent of x over 2 equals plus or minus the fraction with numerator sine x and denominator 1 plus cosine x$

The Half-Angle Formulas

$the sine of x over 2 equals the square root of the fraction with numerator 1 minus cosine x and denominator 2$
$the cosine of x over 2 equals the square root of the fraction with numerator 1 plus cosine x and denominator 2$
$the tangent of x over 2 equals plus or minus the fraction with numerator 1 minus cosine x and denominator sine x or plus or minus the fraction with numerator sine x and denominator 1 plus cosine x$

The sign ± depends on the quadrant x over 2 is in.

Learn

Power-Reducing Formulas

Power-Reducing Formulas

Power-Reducing Formulas

The Power-Reducing Formulas

Half-Angle Formulas

Half-Angle Formulas

Half-Angle Formulas

Simplifying the Tangent Half-Angle Formula

The Half-Angle Formulas

Use Half-Angle Formulas to Find Exact Trigonometric Values

Use Power-Reducing Formulas to Rewrite Trigonometric Expressions

Use Half-Angle Formulas to Solve Trigonometric Equations