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Trigonometry Massoud Malek Right-Angled Triangle The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. opposite a sin A = = . hypothenuse h The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. adjacent b cos A = = . hypothenuse h The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side . a opposite tan A = = . adjacent b Reciprocal Trigonometric functions. The remaining three functions are best defined using the above three functions. The cosecant is the reciprocal of sine: csc A = hypothenuse h = . opposite a Massoud Malek Trigonometry The secant is the reciprocal of cosine: csc A = hypothenuse h = . adjacent b The cotangent reciprocal of tangent: cot A = adjacent b = . opposite a Unit Circle. A unit circle is a circle of radius one centered at the origin (0, 0). 2 Massoud Malek Trigonometry 3 Massoud Malek Trigonometry 4 x2 + y 2 = 1 =⇒ sin2 θ + cos2 θ = 1 By dividing both sides of the above identity by cos2 θ and sin2 θ respectively, we obtain: sin2 θ cos2 θ 1 + = =⇒ tan2 θ + 1 = sec2 θ 2 2 cos θ cos θ cos2 θ 1 sin2 θ cos2 θ + = =⇒ 1 + cot2 θ = csc2 θ 2 2 2 sin θ sin θ sin θ Here are some other identities: sin(a ± b) = sin a cos b ± sin b cos a tan(a ± b) = tan a ± tan b 1 ∓ tan a tan b cos(a ± b) = cos a cos b ∓ sin a sin b cot(a ± b) = 1 ± cot a cot b cot a ∓ cot b cos(a + b) + cos(a − b) cos(a − b) − cos(a + b) sin a sin b = 2 2 sin(a + b) + sin(a − b) sin 2a sin a cos b = sin a cos a = 2 2 1 − cos 2a 1 + cos 2a sin2 a = cos2 a = 2 2 cos a cos b = Limits. sin x =1 x→0 x lim sin x =0 x→∞ x lim Massoud Malek Trigonometry 5 Mac Lauren Series. ∞ X x2n+1 sin x = (−1)n 2n + 1 n=0 ∞ X x2n cos x = (−1)n 2n n=0 Complex forms sinx = ei x − e−i x 2i cos x = ei x + e−i x 2 de Moivre’s formula. (cos x + i sin x)n = cos n x + i sin n x Laws of Sines and Cosines Let T = (a, b, c; A, B, C) be a triangle , where a, b, c are the sides and A, B, C are the angles. Then T satisfies the following conditions: sin A sin B sin C = = a b c Law of Sines : Law of Cosines : c2 = a2 + b2 − 2 a b cos C Derivatives and Integrals. R f (x) sin x cos x tan x cot x sec x csc x f 0 (x) cos x − sin x sec2 x − csc2 x sec x tan x − csc x cot x f (x)dx − cos x sin x ln | sec x| ln | sin x| ln | sec x+tan x| ln | csc x−cot x| Inverse Trigonometric Functions. Let f (x) = sin x, then the inverse function f −1 (x) = arcsin x = sin−1 x is called the arcsine of x . If f (x) is a one to one function in a domain, then f −1 (x) is also a function. Thus sin(arcsin x) = x f or x ∈ [−1, 1] Massoud Malek Trigonometry 6 and arcsin (sin x) = x x ∈ [−π/2, π/2]. whenever Similarly we may define arccos x = cos−1 x, arctan x, etc... f (x) sin−1 x cos−1 x tan−1 x cot−1 x sec−1 x csc−1 x Domain [−1, 1] [−1, 1] (−∞, ∞) (−∞, ∞) |x| ≥ 1 |x| ≥ 1 π 2 (0, π) [0, π] − { π2 } 1 1 + x2 −1 1 + x2 1 x x2 − 1 Range f 0 (x) |x| ≤ √ π 2 [0, π] −1 √ 1 − x2 1 1 − x2 |x| ≤ √ |x| ≤ π 2 −1 x x2 − 1 √ To integrate inverse trigonometric functions, we use integration by parts with v(x) = 1. Z Z sin−1 x dx = x sin−1 x + cos−1 x dx = x cos−1 x − √ 1 − x2 + C √ 1 − x2 + C Z tan−1 x dx = x tan−1 x − 1 ln(1 + x2 ) + C 2 Z cot−1 x dx = x cot−1 x + 1 ln(1 + x2 ) + C 2 Z Z sec−1 x dx = x sec−1 x − ln |x + csc−1 x dx = x csc−1 x + ln |x + √ x2 − 1| + C √ 1 − x2 + C Massoud Malek Trigonometry 7 Identities. sin−1 x + cos−1 x = π , 2 π , 2 1 −1 −1 sin x = csc , x sec−1 x + csc−1 x = sec −1 −1 x = cos 1 , x cos(sin−1 x) = sin(cos−1 x) = sin−1 x = − sin−1 x √ f or |x| ≤ 1. f or |x| ≥ 1. f or |x| ≤ 1. f or |x| ≥ 1. 1 − x2 sec(tan−1 x) = cos−1 x = π − cos−1 x The following table is obtained from the unit circle: √ x2 + 1 tan−1 x = − tan−1 x Massoud Malek Trigonometry 8 x sin x cos x tan x cot x sec x csc x 0 0 1 0 ±∞ 1 ±∞ π/6 1/2 √ 3/2 3/3 √ 3 √ 2 3/3 2 √ √ π/4 π/3 √ √ 2/2 √ 2/2 3/2 1/2 √ 1 1 √ √ 3/3 2 √ 2 3/3 3 2 2 π/2 1 0 ±∞ 0 +∞ 1 −x − sin x cos x − tan x − cot x sec x − csc x π/2 − x cos x sin x cot x tan x csc x sec x π/2 + x cos x − sin x − cot x − tan x − csc x sec x π−x sin x − cos x − tan x − cot x − sec x csc x π+x − sin x − cos x tan x cot x − sec x − csc x Massoud Malek Trigonometry 9 The following table shows how trigonometric functions are related to each other. f (x) sin x sin x sin x p cos x 1−sin2 x cos x tan x cot x √ tan x 1 √ 1−cos2 x √ 2 1 + tan x 1 + cot2 x cos x √ 1 1+tan2 x √ sec x csc x sec2 x − 1 sec x 1 csc x cot x 1 sec x √ 1+cot2 x √ 1−cos2 x tan x p cos x 1−sin2 x tan x 1 cot x √ p 1−sin2 x cos x √ cot x sin x 1−cos2 x 1 tan x cot x √ p 1+tan2 x cot x √ 1+cot2 x sin x 1 sec x p 1−sin2 x csc x 1 sin x 1 cos x 1 √ 1−cos2 x √ 1+tan2 x tan x p 1+cot2 x √ √ csc2 x−1 csc x 1 csc2 x−1 sec2 x−1 √ 1 sec2 x−1 √ sec x √ sec x sec2 x−1 csc2 x−1 csc x csc2 x−1 csc x