Trigonometric Identities

1
Trigonometric Identities
2
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.
3
Examples
Prove that cot A tan A ? sec A cosec A
L.H.S.
R.H.S.
4
Examples
R.H.S.
R.H.S.
5
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?
6
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?
7
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
8
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
9
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
10
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
11
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
12
Examples
L.H.S.
R.H.S.
13
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
14
Double angle formulae
15
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
16
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 ?
17
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 ?
18
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?
19
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 ? )
20
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? )
21
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? )
22
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? )
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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