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DOUBLE-ANGLE, POWER-REDUCING, AND HALF-ANGLE FORMULAS
Introduction
- Another collection of identities called double-angles and half-angles, are acquired from the sum and difference identities in section 2 of this chapter.
- By using the sum and difference identities for both sine and cosine, we are able to compile different types of double-angles and half angles
- First we are going to concentrate on the double angles and examples.
Double-Angles Identities
sin (x + y) = (sin x)(cos y) + (cos x)(sin y) sin (x + x) = (sin x)(cos x) + (cos x)(sin x) (replace y with x) sin 2x = 2 sin x cos x
Double-angle identity for sine.
- There are three types of double-angle identity for cosine, and we use sum identity for cosine, first:
cos (x + y) = (cos x)(cos y) โ (sin x)(sin y) cos (x + x) = (cos x)(cos x) โ (sin x)(sin x) (replace y with x) cos 2x = cos^2 x โ sin 2 x
First double-angle identity for cosine
- use Pythagorean identity to substitute into the first double-angle.
sin^2 x +cos 2 x = 1 cos 2 x = 1 โ sin 2 x
cos 2x = cos^2 x โ sin 2 x cos 2x = (1 โ sin 2 x) โ sin^2 x (substitute) cos 2x = 1 โ 2 sin 2 x
Second double-angle identity for cosine.
Double-Angles Identities (Continued)
- take the Pythagorean equation in this form, sin^2 x = 1 โ cos 2 x and substitute into the First double-angle identity
cos 2x = cos^2 x โ sin 2 x cos 2x = cos^2 x โ (1 โ cos 2 x) cos 2x = cos^2 x โ 1 + cos^2 x cos 2x = 2cos 2 x โ 1
Third double-angle identity for cosine.
Summary of Double-Angles
sin 2x = 2 sin x cos x
cos 2x = cos 2 x โ sin^2 x = 1 โ 2 sin 2 x = 2 cos 2 x โ 1
tan 2x = 2 tan x/1- tan 2 x = 2 cot x/ cot 2 x - = 2/cot x โ tan x
tangent double-angle identity can be accomplished by applying the same methods, instead use the sum identity for tangent, first.
- Note: sin 2x โ 2 sin x; cos 2x โ 2 cos x; tan 2x โ 2 tan x
Half-Angle Identities (Continued)
Cosine
- To get the half-angle identity for cosine, we begin with another double-angle identity for cosine
cos 2x = 2cos 2 x โ 1 cos 2m = 2 cos 2 m โ 1 [replace x with m] cos 2x/2 = 2 cos 2 x/2 -1 [replace m with x/2] cos x = 2 cos^2 x/2 - cos 2 x/2= (1 + cos x)/ 2 [solve for cos (x/2)] โ cos 2 x/2 = โ[(1 + cos x)/ 2 ] c os x/2 = ยฑโ[(1 + cos x)/ 2]
Half-angle identity for cosine
- Again, depending on where the x/2 within the Unit Circle, use the positive and negative sign accordingly.
Tangent
- To obtain half-angle identity for tangent, we use the quotient identity and the half- angle formulas for both cosine and sine:
tan x/2 = (sin x/2)/ (cos x/2) (quotient identity) tan x/2 = ยฑโ [(1 - cos x)/ 2] / ยฑโ [(1 + cos x)/ 2] (half-angle identity) tan x/2 = ยฑโ [(1 - cos x)/ (1 + cos x)] (algebra)
Half-angle identity for tangent
- There are easier equations to the half-angle identity for tangent equation
tan x/2 = sin x/ (1 + cos x) 1 st^ easy equation
tan x/2 = (1 - cos x) /sin x 2 nd^ easy equation.
Summary of Half-Angles
- Sine o sin x/2 = ยฑโ [(1 - cos x)/ 2]
- Cosine
o cos x/2 = ยฑโ [(1 + cos x)/ 2]
Summary of Half-Angles (Continued)
- Tangent o tan x/2 = ยฑโ [(1 - cos x)/ (1 + cos x)] o tan x/2 = sin x/ (1 + cos x) o tan x/2 = (1 - cos x)/ sin x
- Remember, pick the positive and negative sign according to where the x/2 lies.
- Note: sin x/2 โ ยฝ sinx; cos x/2 โ ยฝ cosx; tan x/2 โ ยฝ tanx
Example 2: Find exact value for, tan 30 degrees, without a calculator, and use the half- angle identities (refer to the Unit Circle).
Answer
tan 30 degrees = tan 60 degrees/ 2 = sin 60/ (1 + cos 60) = ( 3 / 2) / (1 +1/ 2) = ( 3 / 2) / (3 / 2) = ( 3 / 2) ร(2 / 3) = 3 / 3
