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Double-Angle and Half-Angle Formulas

Dr .Hayk MelikyanDepartmen of Mathematics and CS

melikyan@nccu.edu

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sin2 = 2 sin cos

cos2 = cos2 – sin2 = 1 – 2sin2 = 2cos2 – 1

tan 2 = 2tan1

2tan

Double-Angle Identities

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Three Forms of the Double-Angle Formula

for cos2

2

2

22

sin212cos

1cos22cos

sincos2cos

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Power-Reducing Formulas

2cos1

2cos1tan

2

2cos1cos

2

2cos1sin

2

2

2

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Example

Write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1.

8

2cos33

8

2cos12cos42

42

2cos12cos21

4

2cos2cos21

2

2cos1

2

2cos1sinsinsin

2

224

xxx

xxxx

xxxxx

Solution

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sinx2 = ±

1 – cos x2

cosx2 = ±

1 + cos x2

tanx2 = ±

1 – cos x1 + cos x =

sin x1 + cos x =

1 – cos xsin x

where the sign is determined by the quadrant in which x2 lies.

Half-Angle Identities

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Find the exact value of cos 112.5°.

Solution Because 112.5° 225°/2, we use the halfangle formula for cos /2 with 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the sign in the halfangle formula.

Text Example

cos112.5 cos225

2

1 cos225

2

1 22

2

2 2

4

2 2

2

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Half-Angle Formulas for:

cos1

sin

2tan

sin

cos1

2tan

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Verifying a Trigonometric Identity

Verify the identity: tan cot 2csc2 2

x xx

sin (1 cos ) sin (1 cos )2csc

(1 cos )(1 cos ) (1 cos )(1 cos )

x x x xx

x x x x

2

2sin2csc

1 cos

xx

x

sin sin2csc

1 cos 1 cos

x xx

x x

2

2sin2csc

sin

xx

x

12 2csc

sinx

x

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The Half-Angle Formulas for Tangent

1 cos

tan Quadrant I or III2 1 cos

1 costan Quadrant II or IV

2 1 cos

1 costan in any quadrant

2 sin

sintan in any quadrant

2 1 cos

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Example

2sin1)cos(sin 2

Verify the following identity:

2sin1cossin22

2

cossin22

2cos1

2

2cos1

coscossin2sin

)cos(sin22

2

Solution

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Product-to-Sum and Sum-to-Product Formulas

Product-to-Sum Formulas

)sin()sin(2

1sincos

)sin()sin(2

1cossin

)cos()cos(2

1coscos

)cos()cos(2

1sinsin

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Example

xx 2cos3cos

Solution

Express the following product as a sum or difference:

)5cos()cos(2

1

)23cos()23cos(2

1

2cos3cos

)cos()cos(2

1coscos

xx

xxxx

xx

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Express each of the following products as a sum or difference. a. sin 8x sin 3x b. sin 4x cos x

Solution The product-to-sum formula that we are using is shown in each of the voice balloons.

a.

sin 8x sin 3x 1/2[cos (8x 3x) cos(8x 3x)] 1/2(cos 5x cos 11x)

sin sin = 1/2 [cos( - ) - cos( + )]

sin cos = 1/2[sin( + ) + sin( - )] b.

sin 4x cos x 1/2[sin (4x x) sin(4x x)] 1/2(sin 5x sin 3x)

Text Example

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Evaluating the Product of a Trigonometric ExpressionDetermine the exact value of the expression 3

sin cos8 8

1sin sin

2 2 4

1 1

12 2

1 3 3sin sin

2 8 8 8 8

1 2 1

2 2

1sin cos sin( ) sin( )

2

2 1

2 2

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Sum-to-Product Formulas

2sin

2sin2coscos

2cos

2cos2coscos

2cos

2sin2sinsin

2cos

2sin2sinsin

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Example

xx 2sin4sin

Express the difference as a product:

xxxx

xxxxxx

3cossin22

6cos

2

2sin2

2

24cos

2

24sin22sin4sin

2cos

2sin2sinsin

Solution

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Example

xx 4sinsin Solution

Express the sum as a product:

2

3cos

2

5sin2

2

4cos

2

4sin24sinsin

2cos

2sin2sinsin

xx

xxxxxx

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Example

2cot

2tan

sinsin

sinsin yxyx

yx

yx

Verify the following identity:

Solution

2cot

2tan

2sin

2cos

2cos

2sin

2cos

2sin2

2cos

2sin2

sinsin

sinsin

yxyxyxyx

yxyx

yxyx

yxyx

yx

yx

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Example

0 0

Express the following as a product and if possible find

the exact value. cos75 cos 15

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Example

Verify that the following is an identity:

sin 3 sintan

cos3 cos

x xx

x x

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• To solve an equation containing a single trigonometric function:

• Isolate the function on one side of the equation.

sinx = a (-1 ≤ a ≤ 1 )

cosx = a (-1 ≤ a ≤ 1 )

tan x = a ( for any real a )

• Solve for the variable.

Equations Involving a Single Trigonometric Function

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y = cos x

x

y

1

–1

y = 0.5

–4 2–2 4

cos x = 0.5 has infinitely many solutions for – < x <

y = cos x

x

y

1

–1

0.5

2

cos x = 0.5 has two solutions for 0 < x < 2

Trigonometric Equations

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Solve the equation: 3 sin x 2 5 sin x 1.

Solution The equation contains a single trigonometric function, sin x.

Step 1 Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

3 sin x 2 5 sin x 1 This is the given equation.

3 sin x 5 sin x 2 5 sin x 5 sin x – 1 Subtract 5 sin x from both sides.

sin x -1/2

Divide both sides by 2 and solve for sin x.

2 sin x 1 Add 2 to both sides.

2 sin x 2 1 Simplify.

Text Example

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Solve the equation: 2 cos2 x cos x 1 0, 0 x 2.

The solutions in the interval [0, 2) are /3, , and 5/3.

Solution The given equation is in quadratic form 2t2 t 1 0 with t cos x. Let us attempt to solve the equation using factoring.

2 cos2 x cos x 1 0 This is the given equation.

(2 cos x 1)(cos x 1) 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1).

cos x 1/2

2 cos x 1 cos x 1 Solve for cos x.

2 cos x 1 0 or cos x 1 0

Set each factor equal to 0.

Text Example

x x 2 x

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Example

cos29cos7

Solve the following equation:

Solution:

n2

5,3,

1cos

9cos9

cos29cos7

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Example

Solve the equation on the interval [0,2)

Solution:

3

3

2tan

3

7

3

6

7

62

3

3

2tan

and

and

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Example

Solve the equation on the interval [0,2)

Solution:03cos2cos2 xx

0

0

1cos3cos

01cos03cos

0)1)(cos3(cos

03cos2cos2

x

xsolutionno

xx

xx

xx

xx

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Example

Solve the equation on the interval [0,2)

Solution:

3

5,

3

2

1cos

1cos2

sincossin2

sin2sin

x

x

x

xxx

xx

xx sin2sin