Trigonometric Identities & Formulas Tutorial Services – Mission del Paso Campus
Reciprocal Identities 1 sin x csc x
1 csc x sin x
cos x
1 sec x
sec x
1 cos x
tan x
1 cot x
cot x
1 tan x
Pythagorean Identities
sinx = cosx tanx
cosx = sinx cotx
Pythagorean Identities in Radical Form
sin x 1 cos2 x
sin x cos x 1 1 tan 2 x sec2 x 1 cot 2 x csc2 x 2
Ratio or Quotient Identities sin x cos x tan x cot x cos x sin x
2
tan x sec 2 x 1
Note: there are only three, basic Pythagorean identities, the other forms
cos x 1 sin 2 x
are the same three identities, just arranged in a different order.
Odd-Even Identities
Confunction Identities
sin x cos x 2
cos x sin x 2
tan x cot x 2
cot x tan x 2
sec x csc x 2
csc x sec x 2
Also called negative angle identities
Sin (-x) = -sin x
Csc (-x) = -csc x
Cos (-x) = cos x
Sec (-x) = sec x
Tan (-x) = -tan x
Cot (-x) = -cot x
Phase Shift = Period =
Sum and Difference Formulas/Identities sin(u v ) sin u cos v cos u sin v sin(u v ) sin u cos v cos u sin v
cos(u v ) cos u cos v sin u sin v cos(u v ) cos u cos v sin u sin v
tan(u v )
tan u tan v 1 tan u tan v
tan(u v )
tan u tan v 1 tan u tan v
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c b
2 b
How to Find Reference Angles Step 1: Determine which quadrant the angle is in Step 2: Use the appropriate formula
Quad I Quad II Quad III Quad IV
= = = =
is the angle itself 180 – θ or π- θ θ – 180 or θ- π 360 – θ or 2π - θ
1
Reciprocal Identities 1 1 sin x csc x csc x sin x
cos x
1 sec x
sec x
1 cos x
tan x
1 cot x
cot x
1 tan x
sinx = cosx tanx
cosx = sinx cotx
Pythagorean Identities in Radical Form
Pythagorean Identities
sin x 1 cos2 x
sin x cos x 1 1 tan 2 x sec2 x 1 cot 2 x csc2 x 2
Ratio or Quotient Identities sin x cos x tan x cot x cos x sin x
2
tan x sec 2 x 1
Note: there are only three, basic Pythagorean identities, the other forms are the same three identities, just arranged in a different order.
Confunction Identities
cos x sin x 2
sin x cos x 2
tan x cot x 2
cot x tan x 2
sec x csc x 2
csc x sec x 2
Odd-Even Identities Also called negative angle identities
Sin (-x) = -sin x
Csc (-x) = -csc x
Cos (-x) = cos x
Sec (-x) = sec x
Tan (-x) = -tan x
Cot (-x) = -cot x
Sum and Difference Formulas - Identities
sin(u v ) sin u cos v cos u sin v sin(u v ) sin u cos v cos u sin v
tan(u v )
tan u tan v 1 tan u tan v
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cos(u v ) cos u cos v sin u sin v cos(u v ) cos u cos v sin u sin v
tan(u v )
tan u tan v 1 tan u tan v
2
The Unit Circle 90° 3
Tan = -
3
3
cot
tan = undefined & cot= 0
tan =
3
cot =
3
3
120°
60° Tan = 1 cot = 1
Tan =- - 1 Cot = -1
135°
45° 2.09
1.57 1.04
150°
30°
2.35 .785 2.61 3
3
Tan =
cot = 3
-
3
tan =
.523
cot =
3
3
3.14 Tan= 0
Tan=0 & cot=undef
Cot=undef
180°
360° 2(3.14 )=
3.66
3 Tan
3
6.28 3
cot =
3
3.925
tan =
5.75
cot = - 3
3
4.186
5.49 4.71
330°
5.23
210° Tan = -1 Cot = -1
Tan = 1 Cot = 1
225°
315°
240°
270°
300°
3
Tan =
3
cot =
3
tan=undefined
3
tan = -
3
cot =
3
Cot = 0
Definition of Trigonometric Functions concerning the Unit Circle
sin θ =
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opp hyp
cos θ =
adj hyp
tan θ =
opp y adj x
y r
csc θ =
hyp opp
r y
x r
sec θ =
hyp adj
r x
cot θ =
adj opp
x y 3
Right Triangle Definitions of Trigonometric Functions Note: sin & cos are complementary angles, so are tan & cot and sec & cos, and the sum of complementary angles is 90 degrees.
C
opp sin θ = hyp
y r
hyp csc θ = opp
r y r
y
Hypotenuse
adj hyp
cos θ =
x r
sec θ =
hyp adj
opposite
r x A
x
B
adjacent
tan θ =
opp y adj x
cot θ =
adj opp
x y
Adjacent = is the side adjacent to the angle in consideration. So if we are considering Angle A, then the adjacent side is CB
Trigonometric Values of Special Angles Degrees 0° 30° 45°
60°
90°
180°
270°
2
3 2
0
6
4
3
0
1 2
2 2
3 2
1
0
-1
cosθ
1
3 2
2 2
1 2
0
-1
0
tanθ
0
3 3
1
0
undefined
Radians sinθ
To Convert Degrees to Radians, Multiply by
To Convert Radians to Degrees, Multiply by
Vocabulary Cotangent Angles Reference Angle
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3
undefined
rad 180deg
180deg rad
- are two angles with the same terminal side - is an acute angle formed by terminal side of angle(α) with x-axis
4
Double Angle Identities
Half Angle Identities 1 cos A A sin 2 2
sin 2 A 2 sin A cos A
Power Reducing Formulas 1 cos 2u sin 2 u 2
cos 2 A cos2 A sin 2 A
cos
A 1 cos A 2 2
cos2 u
1 cos 2u 2
cos 2 A 2 cos2 A 1
tan
A 1 cos A 2 sin A
tan 2 u
1 cos 2u 1 cos 2u
tan
A sin A 2 1 cos A
cos 2 A 1 2 sin 2 A
tan 2 A
2 tan A 1 tan 2 A
Product-to-Sum Formulas 1 sin u sin v cos(u v ) cos(u v ) 2
Sum-to-Product Formulas x y x y sin x sin y 2 sin cos 2 2
cos u cos v
1 cos(u v) cos(u v) 2
x y x y sin x sin y 2 cos sin 2 2
sin u cos v
1 sin(u v) sin(u v) 2
x y x y cos x cos y 2 cos cos 2 2
cos u sin v
1 sin(u v) sin(u v) 2
x y x y cos x cos y 2 sin sin 2 2 Law of Cosines
Law of Sines Solving Oblique Triangles using sine: AAS, ASA, SSA, SSS, SAS
Cosine: SAS, SSS
Standard Form
a sin A
b sin B
c sin C
or
sin A a
sin B b
sin C c
Alternative Form
b2 c2 a 2 cos A 2bc 2 a c2 b2 cos B 2ac 2 a b 2 c2 cosC 2ab
a b c 2bc cos A 2
2
2
b 2 a 2 c 2 2ac cos B c 2 b 2 a 2 2ab cos C Finding the Area of non-90degree Triangles Area of an Oblique Triangle area
1 2
bc sin A
1 2
ab sin C
1 2
ac sin B
Heron’s Formula Step 1: Find “s” Step 2: Use the formula
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s area
a b c 2 s( s a )( s b)( s c)
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