All in one page trigo
1
1) Angles can be measured in degrees and minutes or in radians. ( π radian = 180 0 ) 2) Positive angles are angle measured in the anticlockwise direction from the positive x –axis. 3) Negative angle are angle measured in the clockwise direction from the positive x – axis 4) Quadrants Reference Angles EXAMPLES a) 45 0 b) - 70 0 c) 430 0 4) Six Trigonometric Functions Of Any Angles 5) The signs of the trigonometric functions. I II III IV y x θ r Sin θ = y / r cos θ = x / r tan θ = y / x cosec θ = 1 / sin θ sec θ = 1 / cos θ cot θ = 1 / tan θ All positive Sin θ + Cosec θ + tan θ + Cot θ + Cos θ + Sec θ + S+ A+ T+ C+ TRIGONOMETRIC FUNCTIONS How to determine the value of a trigonometric function of any angle 1) By using Scientific Calculator Eg: a) sin 350 0 (b) kos 3 π 2) Without using calculators a) By using right-angled triangle and trigonometric ratio. Eg: Given cos θ = 3/5 and θ is in quadrant 1. Find the value of sin θ , cosec θ , sec θ ,cot θ b) By using the value of the trigonometric function of the reference angle which is i) a special angle (0 0 , 30 0 , 60 0 , 90 0 ….) ii) a given acute angle c) By using (a) or (b) and trigonometric identities. Special Angles 3O 0 45 0 60 0 Sin cos tan 3 2 1 1 1 1 TRIGONOMETRIC IDENTITIES 1) Complementary angles Identities ) 90 cos( sin θ θ - = o ) 90 tan( cot θ θ - = o ) 90 sin( cos θ θ - = o ) 90 ( cos sec θ θ - = o ec ) 90 cot( tan θ θ - = o ) 90 sec( cos θ θ - = o ec 2) Negative Angles Identities a) Sin (- θ ) = - sin θ b) cos (- θ ) = cos θ c) tan (- θ ) = - tan θ 3) Basic identities a) sin 2 A + cos 2 A = 1 b) 1 + tan 2 A = sec 2 A c) 1 + cot 2 A = cosec 2 A 10) Addition Formulae B A B A B A sin cos cos sin ) sin( ± = ± B A B A B A sin sin cos cos ) cos( = ± B A B A B A tan tan tan tan ) tan( 1 ± = ± 11) Double angle Formulae Sin 2A = 2 sin A cosA Cos 2A =Cos 2 A - sin 2 A = 1 – 2 sin 2 A = 2 Cos 2 A - 1 tan2A = A A 2 tan 1 tan 2 - Half-angle formulae. 2 1 2 2 1 2 2 1 2 1 2 sin cos cos cos sin sin - = = A A A A A = 1 – 2sin 2 A 2 1 = 2 cos 2 A 2 1 - 1 tan A = 2 tan 1 2 tan 2 2 A A - 60 0 5 3 45 0 30 0 0 0 / 360 0 90 0 180 0 270 0 θ θ
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Transcript of All in one page trigo
) sin2 A) sin2 A
1) Angles can be measured in degrees and minutes or in radians. ( π radian = 1800 )
2) Positive angles are angle measured in the anticlockwise direction from the positive x –axis.
3) Negative angle are angle measured in the clockwise direction from the positive x – axis
4) Quadrants Reference Angles
EXAMPLES
a) 450 b) - 700 c) 4300
4) Six Trigonometric Functions Of Any Angles
5) The signs of the trigonometric functions.
III
III IV
y
xθ
r
Sin θ = y / rcos θ = x / rtan θ = y / x
cosec θ = 1 / sin θsec θ = 1 / cos θcot θ = 1 / tan θ
All positiveSin θ +Cosec θ +
tan θ +Cot θ +
Cos θ +Sec θ +
S+ A+
T+ C+
TRIGONOMETRIC FUNCTIONS
How to determine the value of a trigonometric function of any angle
1) By using Scientific Calculator
Eg: a) sin 3500 (b) kos 3
π
2) Without using calculators
a) By using right-angled triangle and trigonometric ratio.
Eg: Given cos θ = 3/5 and θ is in quadrant 1. Find the value of sin θ , cosec θ , sec θ ,cot θ
b) By using the value of the trigonometric function of the reference angle which is i) a special angle (00, 300, 600 , 900….) ii) a given acute angle
c) By using (a) or (b) and trigonometric identities.
Special Angles
3O0 450 600
Sin
cos
tan
32
1 1
1
1
TRIGONOMETRIC IDENTITIES
1) Complementary angles Identities
)90cos(sin θθ −= o )90tan(cot θθ −= o
)90sin(cos θθ −= o )90(cossec θθ −= oec
)90cot(tan θθ −= o )90sec(cos θθ −= oec
2) Negative Angles Identitiesa) Sin (-θ ) = - sinθ
b) cos (-θ ) = cos θ
c) tan (-θ )= - tan θ
3) Basic identities
a) sin2 A + cos2 A = 1
b) 1 + tan2 A = sec2 A
c) 1 + cot2 A = cosec2 A
10) Addition Formulae
BABABA sincoscossin)sin( ±=±BABABA sinsincoscos)cos( =±
BA
BABA
tantan
tantan)tan(
1
±=±
11) Double angle Formulae
Sin 2A = 2 sin A cosA
Cos 2A =Cos2A - sin2A
= 1 – 2 sin2A
= 2 Cos2A - 1
tan2A = A
A2tan1
tan2
−
Half-angle formulae.
2
122
12
2
1
2
12
sincoscos
cossinsin
−=
=
AA
AAA
= 1 – 2sin2 A2
1
= 2 cos2 A2
1- 1
tan A =
2tan1
2tan2
2 A
A
−
600
53
450
300
00 /3600
900
1800
2700
θ θ
