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PPT – Trigonometric Identities PowerPoint presentation | free to download

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Trigonometric Identities

Examples

Prove that (1 cos A)(1 sec A) ?? sin A tan A

L.H.S.

(1 cos A)(1 sec A) 1 sec A cos A Cos A

sec A

1 sec A cos A - 1

sec A cos A

sin A tan A

R.H.S.

Examples

Prove that cot A tan A ? sec A cosec A

L.H.S.

R.H.S.

Examples

R.H.S.

R.H.S.

Solving equations

Solve 2 tan2 x 7 sec x 8 0 for 0 ? x ?

360?

2 (sec2x 1) 7 sec x 8 0

2 sec2x 2 7 sec x 8 0

2 sec2x 7 sec x 6 0

(2 sec x 3)(sec x 2) 0

sec x 3/2 or sec x 2

cos x 2/3 or cos x ½

x 48.2 or x 60

or x 360 48.2 or x 360 - 60

complete solution x 48.2? or 60? or 300? or

311.8?

Solving equations

Solve 2 cos x cot x for 0 ? x ? 360?

2 cos x cos x/ sin x

2 cos x sin x cos x

2 cos x sin x cos x 0

cos x(2 sin x 1) 0

cos x 0 or sin x ½

cos x 0 ? x 90? or 270?

sin x ½ ? x 30? or 330?

complete solution x 30? or 90? or 270? or

30?

Solving equations

Solve 3 cot2 x 10 cot x 3 0 for 0 ? x

? 2?

(3 cot x - 1)(cot x 3) 0

cot x 1/3 or cot x 3

? tan x 3 or tan x 1/3

tan x 3 ? x 1.24c or 4.39c

tan x 1/3 ? x 0.32c or 3.46c

complete solution x 0.32c or 1.24c or

3.46c or 4.39c

Solving equations

Solve 5 cot2 x 2 cosec x 2 0 for 0 ? x

? 2?

5(cosec2 x 1) 2 cosec x 2 0

5cosec2 x 5 2 cosec x 2 0

5cosec2 x 2 cosec x - 3 0

sin x -5/3 not possible or sin x 1 ? x

?/2

Additional formulae

sin (A B) sin A cos B sin B cos A

sin (A - B) sin A cos B - sin B cos A

cos (A B) cos A cos B - sin A sin B

cos (A - B) cos A cos B sin A sin B

Examples

Find the exact value of sin 75?

sin (A B) sin A cos B sin B cos A

sin (30 45) sin 30 cos 45 sin 45 cos 30

Examples

Express cos (x ?/3) in terms of cos x and sin x

cos (A B) cos A cos B - sin A sin B

cos (x ?/3) cos x cos ?/3 - sin ?/3 sin x

Examples

L.H.S.

R.H.S.

Double angle formulae

sin (A B) sin A cos B sin B cos A

sin (A A) sin A cos A sin A cos A

sin 2A 2 sin A cos A

cos (A B) cos A cos B - sin A sin B

cos (A A) cos A cos A- sin A sin A

cos (A A) cos2A - sin2A

cos 2A cos2A - sin2A

cos 2A 2cos2A - 1

cos 2A 1 2sin2A

Double angle formulae

Examples

Given that cos A 2/3, find the exact value of

cos 2A.

cos 2A 2cos2A - 1

Given that sin A ¼ , find the exact value of

sin 2A.

sin 2A 2 sin A cos A

Solving equations

Solve cos 2A 3 4 cos A 0 for 0 ? x ? 2?

2 cos2A - 1 3 4 cos A 0

2 cos2A 4 cos A 2 0

cos2A 2 cos A 1 0

cos2A 2 cos A 1 0

(cos A 1)2 0

cos A - 1

? A ?

Solving equations

Solve sin 2A sin A for - ? ? x ? ?

2sin A cos A sin A

2 sin A cos A sin A 0

sin A(2 cos A 1) 0

? sin A 0 or cos A ½

sin A 0 ? A - ? or 0 or ?

cos A ½ ? A - ?/3 or ?/3

Complete solution A - ? or - ?/3 or 0 or ?/3

or ?

Solving equations

Solve tan 2A 5 tan A 0 for 0? x ? 2?

tan A 0 ? A 0 or ? or 2?

7 5tan2 A 0? tan A ? ?7/5 ? A 0.97 ,

2.27, 4.01 or 5.41c

Complete solution A 0.97 , 2.27, 4.01, 5.41c 0,

? or 2?

Harmonic form

If a and b are positive

a sin x b cos x can be written in the form R

sin( x ? )

a sin x - b cos x can be written in the form R

sin( x - ? )

a cos x b sin x can be written in the form R

cos( x - ? )

a cos x - b sin x can be written in the form R

cos( x ? )

Examples

Express 3 cos x 4 sin x in the form R cos( x -

? )

R cos( x - ? ) R cos x cos ? R sin x sin ?

3 cos x 4 sin x R cos x cos ? R sin x sin ?

R cos ? 3 1 R sin ? 4 2

12 22 R2 sin2 x R2 cos2 x 32 42

R2(sin2 x cos2 x ) 32 42

R2 32 42 25 ? R 5

2 ? 1 tan ? 4/3 ? ? 53.1

3 cos x 4 sin x 5 cos( x 53.1? )

Examples

Express 12 cos x 5 sin x in the form R sin( x

? )

R sin( x ? ) R sin x cos ? R cos x sin ?

12 cos x 5 sin x R sin x cos ? R cos x sin ?

R cos ? 12 1 R sin ? 5 2

12 22 R2 cos2 x R2 sin2 x 122 52

R2(cos2 x sin2 x ) 122 52

R2 122 52 169 ? R 13

2 ? 1 tan ? 5/12 ? ? 22.6

12 cos x 5 sin x 13 sin( x 22.6? )

Examples

Express cos x - ?3 sin x in the form R cos( x

? )

R cos( x ? ) R cos x cos ? - R sin x sin ?

cos x - ?3 sin x R cos x cos ? - R sin x sin ?

R cos ? 1 1 R sin ? ?3

2

12 22 R2 cos2 x R2 sin2 x 12

(?3 ) 2

R2(cos2 x sin2 x ) 12 3

R2 1 3 4 ? R 2

2 ? 1 tan ? ?3 ? ? 60

cos x ?3 sin x 2 cos( x 60? )

Solving equations

Solve 7 sin x 3 cos x 6 for 0? x ? 2?

R sin( x ? ) R sin x cos ? R cos x sin ?

7 sin x 3 cos x R sin x cos ? R cos x sin ?

R cos ? 7 1 R sin ? 3 2

R2 72 32 ? R 7.62

2 ? 1 tan ? 3/7 ? ? 0.405c

(Radians)

7 sin x 3 cos x 7.62 sin( x 0.405)

7.62 sin( x 0.405 ) 6 ? x 0.405

sin-1(6/7.62)

x 0.405 0.907 or 2.235

x 0.502c or 1.830c

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For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

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