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)
5