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Summary of Trigonometric Facts Formulas Involving Radian Angular Measure 180 s v 1 rad = π deg = = r r π 1 deg = 180 rad Trigonometric Function Definitions b opp = R hyp R hyp cosecant x = csc x = b = opp sine x = sin x = A sin A cos A tan A 0˚ 0 0 30˚ π/6 1/ 2 1 3/2 3/ 3 0 45˚ π/4 a adj = R hyp R hyp secant x = sec x = = adj a b opp = a adj a adj cotangent x = cot x = = opp b tangent x = tan x = y Q II QI sin x > 0 All are positive. csc x > 0 Others are negative. 90˚ π/2 2/2 3/ 2 1 0 2 / 2 1/2 1 3 undef'd y 1 .5 R = a2 + b2 cosine x = cos x = 60˚ π/3 A = 12 r2 Q III Q IV tan x > 0 cos x > 0 cot x > 0 sec x > 0 Others are negative. Others are negative. y y = sin x Amplit ude = 1 π/2 π y = cos x π 2π x 3π/2 3π/2 2π x π/2 –.5 Amplit ude = 1 –1 Period = 2π Period = 2π y y y = cot x y = tan x –π/2 –π/4 1 π/4 1 x π/2 3π/4 π/2 π/4 π x Period = π Period = π y y y = csc x 1 π/2 π Period = 2π 3π/2 y = sec x 2π x 1 π/2 π Period = 2π 3π/2 2π x csc x = Reciprocal Identities 1 sec x = cos cot x = tan1 x x 1 sin x Tangent and Cotangent Identities sin x = tan x cos x = cot x cos x sin x Sum and Difference Formulas sin(x ± y) = sin x cos y ± cos x sin y cos(x ± y) = cos x cos y m sin x sin y tan( x ± y) = Pythagorean Identities sin2 x + cos2 x = 1 1 + tan2 x = sec 2 x 1 + cot2 x = csc 2 x tan x ± tan y 1 m tan x tan y Double Angle Fomulas sin 2x = 2 sin x cos x cos 2x = cos2x – sin2x = 2 cos2x–1 =1–2 sin2x 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 sin(− x) = − sin x csc(− x) = − csc x cos(− x) = cos x sec(− x) = sec x tan(− x) = − tan x cot(− x) = − cot x tan 2x = 2tan x 1–tan 2 x Half Angle Formulas sin x2 sin sin =± 1–cos x 2 cos = ± x 2 x cos y = 12 [(sin(x + y) x sin y = 12 [cos(x – y) 1+cos x 2 x 1−cos x sin x tan 2x = ± 1−cos 1+cos x = sin x = 1+ cos x Product to Sum or Difference Formulas + sin( x – y)] cos x sin y = 12 [sin(x + y) – sin( x – y)] – cos( x + y)] cos x cos y = 12 [cos(x + y) + cos( x – y)] Inverse Trigonometric Functions arcsin x or sin-1x ∈ ,2 arccsc x or csc–1x ∈ −2 ,0 U 0, 2 arcsec x or sec–1x ∈ 0, 2 U 2 , arccos x or cos–1x ∈ [0,π] [ − 2 ( arctan x or tan-1x ∈ −2 , 2 ] ) [ )( ] [ )( ] arccot x or cot–1x ∈ ( −2 ,0) U(0, 2 ] Law of Sines a b c = = sin A sinB sinC Law of Cosines = b 2 + c 2 – 2bc cos A = a 2 + c 2 – 2ac cos B c = a 2 + b 2 – 2ab cos C a2 b2 2 Area of a Triangle area = 12 area = 12 area = 12 bc sin A ac sin B ab sin C Heron's Formula area = s(s – a)(s – b)(s – c) , a+b +c where s = 2 Sum of Sine and Cosine with Same Period a sin cx + b cos cx = A sin(cx + ), b a where A = a 2 + b 2 , sin = , and cos = a2 + b2 a2 + b2