Summary of Trigonometric Facts

Formulas Involving Radian Angular Measure

1 deg = π

180 rad 1 rad = 180π deg

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θ = sr

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ω = vr

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A = 12 θ r2

Trigonometric Function Definitions

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r = x 2 + y 2( )

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sine θ = sinθ =yr

=opphyp

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cosine θ = cosθ =xr

=adjhyp

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tangent θ = tanθ =yx

=oppadj

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cosecant θ = cscθ =ry

=hypopp

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secant θ = secθ =rx

=hypadj

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cotangent θ = cotθ =xy

=adjopp

Trig Function Values at Special Angles

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0°

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30°

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45°

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60°

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90°

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0

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π / 6

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π / 4

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π / 3

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π / 2A

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sin A

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cos A

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tan A

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0

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1 / 2

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2 / 2

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3 / 2

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1

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1

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3 / 2

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2 / 2

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1 / 2

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0

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0

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3 / 3

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1

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3 undef’d

Signs of the Trig Functions in the Quadrants

x

yQ I

All are positive

Q IIsin A and csc A are

positive; others are negative.

Q IIItan A and cot A are positive; others are

negative.

Q IVcos A and sec A are positive; others are

negative.

π/2 π 3π/22π

.5

1

y = sin x

x

y

Period = 2π

Amplit ude = 1

–1

π/2π 3π/2 2π

x

y

–.5

y = cos x

Period = 2π

Amplit ude = 1

–π/4–π/2π/4 π/2

x

y

1

Period = π

y = tan x

π/4 π/23π/4 π1

y

x

Period = π

y = cot x

π/2 π3π/2 2π

x

y

1

Period = 2π

y = csc x

y

xπ/2 π 3π/2 2π1

Period = 2π

y = sec x

Reciprocal Identities

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csc x = 1sin x

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sec x = 1cos x

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cot x = 1tan x

Tangent and Cotangent Identities

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sin xcos x = tan x

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cos xsin x = cot x

Pythagorean Identities

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sin2 x + cos2 x =1

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1 + tan2 x = sec2 x

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1 + cot2 x = csc2 x

Sum and Difference Formulas

sin(x ± y) = sin x cos y ± cos x sin y

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cos(x ± y) = cos x cos y sin x sin y

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tan(x ± y) = tanx ± tan y1 tanx tan y

Double Angle Fomulas

sin 2x = 2 sin x cos x Cofunction Identities

sin x = cos(π/2 – x) cos x = sin(π/2 – x) tan x = cot(π/2 – x) csc x = sec(π/2 – x) sec x = csc(π/2 – x) cot x = tan(π/2 – x)

Even-Odd Identities

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sin(−x) = − sin x

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csc(−x) = −csc x

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cos(−x) = cos x

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sec(−x) = sec x

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tan(− x) = − tan x

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cot(− x) = −cot x

cos 2x = cos2x – sin2x = 2 cos2x–1 =1–2 sin2x

tan 2x =

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2tan x1 – tan2 x

Half Angle Formulas

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sin x2 = ± 1–cos x2

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cos x2 = ± 1+cos x2

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tan x2 = ± 1−cos x1+cos x = 1−cos x

sinx = sin x1+ cos x

Product to Sum or Difference Formulas

sin x cos y =

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12 (sin(x + y) + sin(x – y)[ ] cos x sin y =

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12 sin(x + y) – sin(x – y)[ ]

sin x sin y =

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12 cos(x – y) – cos(x + y)[ ] cos x cos y =

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12 cos(x + y) + cos(x – y)[ ]

Inverse Trigonometric Functions

arcsin x or sin-1x ∈

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−π2 ,

π2[ ] arccsc x or csc–1x ∈

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−π2 ,0)[ 0, π2](

arccos x or cos–1x ∈ [0,π] arcsec x or sec–1x ∈

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0,π2 )[ π2 ,π ](

arctan x or tan-1x ∈

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−π2 ,

π2( ) arccot x or cot–1x ∈

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−π2 ,0) 0,π2 ]((

Law of Sines

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asinA =

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bsinB =

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csinC

Law of Cosines

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a2 = b2 +c 2 – 2bc cos A

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b2 = a2 +c 2 – 2ac cos B

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c 2 = a2 +b2 – 2ab cos C

Area of a Triangle area = 12 bc sin A

area = 12 ac sin B

area = 12 ab sin C

Heron's Formula

area =

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s(s – a)(s – b)(s – c) ,

where

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s = a + b + c2

Sum of a Sine and a Cosine with the Same Period

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a sin cx + b cos cx = A sin(cx + φ), where

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A = a2 + b2 ,

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sinφ =b

a2 + b2,

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cosφ =a

a2 + b2