Trigonometric Identities & Formulas PDF

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Trigonometric Identities & Formulas Tutorial Services – Mission del Paso Campus Reciprocal Identities Ratio or Quotient Identities 1 1 sin x cos x sin x  csc x  tan x  cot x  csc x sin x cos x sin x 1 1 cos x  sec x  sinx = cosx tanx cosx = sinx cotx sec x cos x 1 1 tan x  cot x  cot x t...

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Trigonometric Identities & Formulas Tutorial Services – Mission del Paso Campus

Reciprocal Identities 1 sin x  csc x

1 csc x  sin x

Ratio or Quotient Identities sin x cos x tan x  cot x  cos x sin x sinx = cosx tanx

cos x 

1 sec x

sec x 

1 cos x

tan x 

1 cot x

cot x 

1 tan x

Pythagorean Identities

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

cosx = sinx cotx

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.

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 

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

Phase Shift = Period =

sin(u  v )  sin u cos v  cos u sin v sin(u  v )  sin u cos v  cos u sin v

Sum and Difference Formulas/Identities 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  v ) 

tan u  tan v 1  tan u tan v

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

tan u  tan v 1  tan u tan v

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2 b

c b

1

Reciprocal Identities 1 1 sin x  csc x  csc x sin x

Ratio or Quotient Identities sin x cos x tan x  cot x  cos x sin x

cos x 

1 sec x

sec x 

1 cos x

sinx = cosx tanx

tan x 

1 cot x

cot x 

1 tan x

Pythagorean Identities

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

cosx = sinx cotx

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 

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

  csc  x  sec x 2 

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° Tan = -

3

cot 

3

3

tan =

tan = undefined & cot= 0

cot =

3

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

Tan = 

3

3

cot =

-

tan =

.523

3

cot =

3

3

3

3.14 Tan= 0

Tan=0 & cot=undef

Cot=undef

180°

360° 2(3.14 )=

3.66

6.28 

3 cot =

Tan

3.925

3

tan =

5.75

3

3

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 =

tan=undefined

tan = -

3

cot =

3



3 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.

opp sin θ =  hyp

C

hyp csc θ =  opp

y r

r y r

adj  hyp

cos θ =

x r

sec θ =

hyp  adj

opposite

r x A

tan θ =

opp y  adj x

cot θ =

y

Hypotenuse

adj  opp

x

B

adjacent

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°

6

4

3

2

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

0

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

180°

270°



3 2

 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

sin 2 A  2 sin A cos A

Half Angle Identities 1  cos A A sin   2 2

cos 2 A  cos2 A  sin 2 A

cos

cos 2 A  2 cos2 A  1

tan

tan 2 A 

tan

cos 2 A  1  2 sin 2 A

2 tan A 1  tan 2 A

A 1  cos A  2 2

cos2 u 

1  cos 2u 2

A 1  cos A  2 sin A

tan 2 u 

1  cos 2u 1  cos 2u

A sin A  2 1  cos A

Product-to-Sum Formulas 1 sin u sin v  cos(u  v )  cos(u  v ) 2

cos u cos v 

1 cos(u  v)  cos(u  v) 2

sin u cos v 

1 sin(u  v)  sin(u  v) 2

cos u sin v 

1 sin(u  v)  sin(u  v) 2

Power Reducing Formulas 1  cos 2u sin 2 u  2

Sum-to-Product Formulas  x  y  x  y sin x  sin y  2 sin   cos  2   2 

 x  y  x  y sin x  sin y  2 cos   sin  2   2 

 x  y  x  y cos x  cos y  2 cos  cos   2   2   x  y  x  y cos x  cos y   2 sin   sin  2   2 

Law of Sines

Law of Cosines

Solving Oblique Triangles using sine: AAS, ASA, SSA, SSS, SAS

a sin A



b sin B



c sin C

or

sin A a



sin B b



Cosine: SAS, SSS

b2  c2  a 2 cos A  2bc 2 a  c2  b2 cos B  2ac 2 a  b 2  c2 cosC  2ab

Standard Form

sin C c

Alternative Form

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...