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Trigonometric Functions of right triangle The Trigonometric Ratios opposite b hypotenuse a c θ adjacent sin θ = adj. c opp. b = cos θ = = hyp. a hyp. a tan θ = opp . b adj . c = cot θ = = adj . c opp. b sec θ = hyp. a hyp. a = csc θ = = adj . c opp. b Exp: Find the six trigonometric ratios of the angle in the following figure. 3 2 θ sin θ = tan θ = sec θ = 5 5 2 cos θ = 3 3 2 5 3 5 cot θ = csc θ = 5 2 3 2 Note: cot θ = 1 1 1 sec θ = csc θ = tan θ cos θ sin θ Special Triangles θ in degrees θ in radiuss sin θ cos θ tan θ 30o π 1 2 3 2 3 3 45o π 2 2 1 1 60o π 3 2 2 2 1 2 3 3 3 6 4 3 cot θ 3 sec θ csc θ 2 3 3 2 2 2 2 2 3 3 Fundamental Identities 1. tan θ = sin θ cos 2 θ 2. cot θ = cos θ sin θ 3. sin 2 θ + cos 2 θ = 1 4. tan 2 θ + 1 = sec 2 θ 5. 1 + cot 2 θ = csc 2 θ Trigonometric functions of Real numbers Def: Let t be any real number and let P(x, y) be the terminal point on the unit circle determined by θ . We define sin ϑ = y cos θ = x tan θ = csc θ = y ( x ≠ 0) x 1 x 1 sec θ = ( x ≠ 0) cot θ = ( y ≠ 0) x y y Even‐odd Properties sin( −t ) = − sin t cos( − t ) = cos t tan( −t ) = − tan t csc( −t ) = − csc t sec( −t ) = sec t cot( − t ) = − cot t Trigonometric Graphs The functions sine and cosine have period 2π: The functions tangent and cotangent have period π: The functions cosecant and secant have period 2π: Addition and Subtraction Formulas (Trigonometric Function) Formulas of sine: sin( s + t ) = sin s cos t + cos s sin t sin( s − t ) = sin s cos t − cos s sin t Formulas of cosine: cos( s + t ) = cos s cos t − sin s sin t cos( s − t ) = cos s cos t + sin s sin t Exp: If f ( x ) = sin x , show that f ( x + h) − f ( x) cosh − 1 sinh + cos x = sin x h h h Sol: f ( x + h) − f ( x ) sin( x + h) − sin x sin x cosh + cos x sinh − sin x sin x (cosh − 1) + cos x sinh = = = h h h h cosh − 1 sinh = sin x ( ) + cos x h h Exercise: Let g ( x ) = cos x , show that f ( x + h) − f ( x) 1 − cosh sinh = − cos x ( ) − sin x ( ) h h h Formulas for tangent: tan( s + t ) = tan s + tan t 1 − tan s tan t tan( s − t ) = tan s − tan t 1 + tan s tan t Double‐Angles Formulas sin 2 x = 2 sin x cos x cos 2 x = cos 2 x − sin 2 x = 1 − 2 sin 2 x = 2 cos 2 x − 1 tan 2 x = 2 tan x 1 − tan 2 x Formulas for Lowering Powers sin 2 x = 1 − cos 2 x 1 + cos 2 x 2 cos x = 2 2 tan 2 x = 1 − cos 2 x 1 + cos 2 x Exp: Express sin 2 x cos 2 x in terms of the first power of cosine. Sol: sin 2 x cos 2 x = 1 − cos 2 x 1 + cos 2 x 1 1 1 1 + cos 4 x 1 cos 4 x )= − ⋅ = (1 − cos 2 2 x ) = − ( 2 2 4 4 4 2 8 8 Product‐Sum formulas sin u cos v = 1 [sin( u + v ) + sin( u − v )] 2 cos u cos v = 1 [cos( u + v ) + cos( u − v )] 2 sin u sin v = 1 [cos(u − v) − cos(u + v)] 2 Exp: Express sin 3 x sin 5 x as a sum of trigonometric functions. Sol: sin 3 x sin 5 x = 1 [cos( −2 x ) − cos(8 x )] = 1 (cos 2 x − cos 8 x ) 2 2 Exercise: Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. 1. sin 4 x = 3 1 cos 4 x − cos 2 x + 8 2 8 3 1 cos 4 x 2. cos 4 x = + cos 2 x + 8 2 8 Exercise: Write the product as a sum 1. sin 2 x cos 3 x = 1 1 (sin 5 x + sin( − x )) = (sin 5 x − sin x ) 2 2 2. cos 5 x cos 3 x = 1 (cos 8 x + cos 2 x ) 2 One‐to‐One Functions Def: A function with domain A is called a one‐to‐one function if no two elements of A have the same image, that is, f ( x1 ) ≠ f ( x 2 ) whenever x1 ≠ x 2 . Horizontal Line Test A function is one‐to‐one if and only if no horizontal line intersects its graph more than once. The inverse of a Function Def: Let f be a one‐to‐one function with domain A and range B. Then its inverse function f −1 has domain B and range A and is defined by f −1 ( y ) = x ⇔ f ( x) = y for any y ← B . −1 Note: f ( x) ≠ 1 1 = ( f ( x)) −1 f ( x) f ( x) Exp: If f (1) = 5 , f (3) = 7 and f (8) = − 10 , find f Sol: f −1 (5) = 1, f −1 (7) = 3, f −1 −1 (5), f −1 (7), f −1 (−10). (−10) = 8 . Inverse function Property Let f be a one‐to‐one function with domain A and range B. The inverse function f −1 satisfies f −1 ( f ( x)) = x f ( f −1 ( x)) = x ∀x ∈ A ∀x ∈ B How to find the Inverse function 1. Write y = f (x) . 2. Solves this equation for x in terms of y (if possible) 3. Interchange x and y. The resulting equation is y = f Exp: Find the inverse of the function f ( x) = 3x − 2 Sol: 1st. y = 3x − 2 2nd. 3x = y + 2 x = 3rd. y = Ans: f −1 x+2 3 ( x) = x+2 3 y+2 3 −1 ( x) Exp: Let f ( x ) = x − 2 . Find f Sol: 1st. y = x − 2 −1 ( x) y ≥0 2nd. y 2 = x − 2 x = 2 + y 2 3rd. y = 2 + x 2 x ≥ 0 Thus, f −1 ( x) = x 2 + 2 x ≥ 0 Exercise: Find the inverse function of f. 1. f ( x) = 2 x + 1 Sol : f −1 ( x) = 2. f ( x ) = 1 Sol : f x+2 3. f ( x) = 4 − x 2 −1 x −1 2 ( x) = 1 − 2x ( x ≠ −2 ) x x ≥ 0 Sol : f −1 ( x) = 4 − x Inverse Trigonometric Functions The inverse sine function is the function sin −1 with domain [− 1,1] and range ⎡ π π⎤ −1 ⎢− 2 , 2 ⎥ defined by sin x = y ⇔ sin y = x . ⎦ ⎣ The inverse sine function is also called arcsine, denoted by arcsin. The inverse cosine function is the function cos −1 with domain [− 1,1] and range [0, π ] defined by cos −1 x = y ⇔ cos y = x . The inverse cosine function is called arccosine, denoted by arcos. π 1 π 1 Exp: (1) sin −1 ( ) = sin −1 (− ) = − 2 6 2 6 (2) cos −1 ( 3 π π ) = cos −1 0 = 2 6 2 The inverse tangent function is the function tan −1 with domain R and range ⎡ π π⎤ −1 ⎢− 2 , 2 ⎥ defined by tan x = y ⇔ tan y = x . ⎦ ⎣ The inverse tangent function is also called arctangent, denoted by arctan. Exp: tan −1 1 = π π tan −1 3 = 4 3 Exercise: (1) cos −1 1 π = 2 3 3 π = 2 3 −1 (3) tan 0 = 0 (2) sin −1