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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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