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1 6. Trigonometry RECAP For all angles θ the three trigonometric ratios are defined by :sin y r cos x r tan y x (where x 0 ) There are three more trigonometric functions – secant, cosecant and cotangent. They are defined by :- sec 1 cos cosec 1 sin cot 1 tan These functions all have values where they are undefined – this gives places where the graphs have vertical asymptotes. Note that each function is the reciprocal of one of the basic trigonometric functions – and will therefore have similar properties. y = sec x is periodic with period 2π, it is symmetrical about the y axis and has vertical asymptotes at all values where cos x = 0. (i.e. at x = 90 ± 180n° or n ) 2 y = cosec x is periodic with period 2π, it has rotational symmetry about the origin and has vertical asymptotes at all values where sin x = 0. (i.e. at x = 180n° or n ) y = cot x is periodic with period π and has vertical asymptotes at all values where sin x = 0. (i.e. at x = 180n° or n ) Also just as so sin θ = cos (90 – θ ) and cos θ = sin (90 – θ ) sec θ = cosec (90 - θ ) and cosec θ = sec (90 - θ ) So the graphs of y = sec x and y = cosec x are obtained from each other by a horizontal translation of 90° or π rads. 2 THE GRAPHS OF SEC, COSEC AND COT 3 SPECIAL ANGLES We have found the exact trigonometric ratios of angle 30°, 45° and 60°. 0° SINE 0 COSINE 1 TANGENT 0 30° 1 2 3 2 1 3 45° 1 2 1 2 60° 3 2 1 2 90° 1 3 - We can use these to find exact values of common angles for sec, cosec and cot. EXAMPLE Find the exact value of the following cot 315° = sec 210° = cosec 300° = cot2 (-60°) = cosec3 (-225°) = 1 0 4 SIMPLIFYING EXPRESSIONS & SOLVING EQUATIONS Usually questions involving sec, cosec or cot are best attempted by changing them into sin, cos or tan. EXAMPLE Prove the identity cos 2 cosec cosec sin EXAMPLE Solve the equations below for the range given. 3 a) sec 3 b) cosec 3cot 2 2 2 2 5 TRIGONOMETRIC IDENTITIES We have already seen two identities :tan 1. sin cos Hence cot cos sin sin 2 cos 2 1 2. Hence sec2 1 tan 2 And cosec2 1 cot 2 PROOF sin 2 cos 2 1 sin 2 cos 2 1 EXAMPLE Prove the following identities :sec2 cot 2 cosec2 tan 2 cosec4 cot 4 1 cos 2 sin 2 6 EXAMPLE Solve the equations in the given range :a) 2 tan 2 8 7 sec 0 2 b) 2tan 6cot 7 EXAMPLE Obtain an equation in x and y only, by eliminating θ. a) x 4sec , y 3cosec b) x 2 5sec , y 3 5 tan 7 THE INVERSE TRIGONOMETRIC FUNCTIONS For any one-to-one function the graph of its inverse can be obtained by reflecting the graph in the line y = x. In order for the 3 trigonometric functions to be one-to-one we must restrict their domain (xvalues). The notations used for these inverse functions are arcsin x, arccos x and arctan x although sin-1 x, cos-1 x and tan-1 x are also used. x , 1 sin x 1 y = sin x y = arcsin x 1 x 1, arcsin x 2 2 2 2 y = cos x 0 x , 1 cos x 1 y = tan x 2 x 2 , tan x y = arccos x y = arctan x 1 x 1, 0 arccos x x , 2 arctan x 2