Math Handout (Trigonometry) Trig Formulas Web Page PDF

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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

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  cos x  1 1  tan 2 x  sec 2 x 1  cot 2 x  csc 2 x 2

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

sin x   1  cos 2 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.

Confunction Identities

Odd-Even Identities Also called negative angle 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 

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

Pythagorean Identities

sinx = cosx tanx

cosx = sinx cotx

Pythagorean Identities in Radical Form

sin x  cos x  1 1  tan 2 x  sec 2 x 1  cot 2 x  csc 2 x 2

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

sin x   1  cos 2 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

Odd-Even Identities Also called negative angle 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

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 usin 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 )  cosu cosv  sinu sinv cos(u  v )  cosu cosv  sinu sinv

tan(u  v ) 

tan u  tan v 1  tan u tan v

2

The Unit Circle 90° 3

Tan = -

3

3

cot 

tan =

tan = undefined & cot= 0

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

Tan = 

cot =

-

tan =

.523

3

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

3

cot =

Tan

3.925

3

5.75



tan =

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° Tan =

3

cot =

270°

300° 3

3

tan=undefined

tan = -

3

cot =



3

3

Cot = 0

Definition of Trigonometric Functions concerning the Unit Circle

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sin θ =

opp  hyp

y r

csc θ =

hyp  opp

r y

cos θ =

adj  hyp

x r

sec θ =

hyp  adj

r x

tan θ =

opp y  adj 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

cos θ =

adj  hyp

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°



3 2









0

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

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 A 1  cos 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  cos 2 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  tan2 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 sin x  sin y  2 cos   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 Sines

Law of Cosines

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

Cosine: SAS, SSS

Alternative Form

Standard Form

a sin A



b sin B



c sin C

or

sin A a



sin B b



sin C c

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

a  b  c  2bc cos A 2

2

2

b2  a2  c2  2 accos B c2  b2  a2  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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