Summary of Trigonometric Facts Formulas Involving Radian Angular Measure 1 deg = π 180 rad 1 rad = 180 π deg θ = s r ω = v r A = 1 2 θ r 2 Trigonometric Function Definitions r = x 2 + y 2 ( ) sine θ = sinθ = y r = opp hyp cosine θ = cosθ = x r = adj hyp tangent θ = tanθ = y x = opp adj cosecant θ = csc θ = r y = hyp opp secant θ = sec θ = r x = hyp adj cotangent θ = cot θ = x y = adj opp Trig Function Values at Special Angles 0 ° 30 ° 45 ° 60 ° 90 ° 0 π /6 π /4 π /3 π /2 A sin A cos A tan A 0 1/2 2 /2 3 /2 1 1 3 /2 2 /2 1/2 0 0 3 /3 1 3 undef’d Signs of the Trig Functions in the Quadrants x y Q I All are positive Q II sin A and csc A are positive; others are negative. Q III tan A and cot A are positive; others are negative. Q IV cos A and sec A are positive; others are negative. π/2 π 3π /2 2π .5 1 y = sin x x y Period = 2π Amplit ude = 1 –1 π/2 π 3π/2 2π x y –.5 y = cos x Period = 2π Amplit ude = 1 –π/4 –π/2 π/4 π/2 x y 1 Period = π y = tan x π/4 π/2 3π/4 π 1 y x Period = π y = cot x π/2 π 3π/2 2π x y 1 Period = 2π y = csc x y x π/2 π 3π/2 2π 1 Period = 2π y = sec x
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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)