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More trigonometric quantities… Sec, Cosec and Cot Secant, cosecant and cotangent, almost always written as sec, cosec and cot are trigonometric functions like sin, cos and tan. sec x = . cot x = 1 cos x 1 tan x cosec x = 1 sin x = cos x sin x Trigonometry Identities The first identity can be derived from Pythagoras"s theorem sin²x + cos²x = 1 . By dividing each of these terms by sin²x, we can derive a second identity: 1 + cot²x = cosec²x By dividing (*) by cos²x, we arrive at the third (and final) identity: tan²x + 1 = sec²x Compound angle formulae sin(A + B) = sinAcosB + cosAsinB cos(A + B) = cosAcosB - sinAsinB tan(A + B) = tanA + tanB 1 – tanAtanB sin(A - B) = sinAcosB - cosAsinB cos(A - B) = cosAcosB + sinAsinB tan(A - B) = tanA - tanB 1 + tanAtanB sin(2A) = sinAcosA + cosAsinA ie sin2A = 2sinAcosA cos2A = cos2A - sin2A ie cos2A = 2cos2A – 1 tan2A = 2tanA 1 - tan2A cos(A + B) + cos(A - B) = 2cosAcosB cos(A - B) – cos (A + B) = 2sinAsinB sin x + siny sin x - siny cosx + cosy cosx – cosy = 2sin½(x+y)cos½(x-y) = 2cos½(x+y)sin½(x-y) = 2cos½(x+y)cos½(x-y) = -2sin½(x+y)sin½(x-y) General solution for sinӨ = y ( where y is the instantaneous value) If sin Ө = 0.5, Ө = 30, 390, 750 or 150, 510, 870 General solution for sin Ө = 0.5 is ( Ө = n360 + 30) or ((2n+1)180 - 30) General solution for Cos Ө = y If cos Ө = 0.5, Ө = 60, 300, 420 , 660 General solution is Ө= (n360 +60) or (n360 - 60) ie Ө=(n360 + 60) General solution for Tan Ө = y If tan Ө = 1.0, Ө = 45, 225, 405, 585 General solution is Ө = (n180 + 45) rcos(Ө + ), rsin(Ө + )….. forms An expression in the form: acos Ө + bsinӨ can be rewritten in the form rcos(Ө + ) Now rcos(Ө + rcos cosӨ - rsin sinӨ So rcos =a and - rsin = b so tan = -b/a and r2cos2 =a2 and r2sin2 = b2 Hence r2sin2 + r2cos2 = a2 + b2 , ie r2 = a2 + b2 Examples (1) Find all the values, in degrees and radians, that satisfy the following identities sinx = 0.7, cos x =0.7, tanx = 2.1, cotx = 10 secx = 1.5, cosecx =1.5 (2) Express the following as products of 2 trig ratios (a) sin5x + sinx (b) cos5x + cos3x (c) sin5x - sin2x (d) cos5x – cos7x (3) Express the following as the sum of 2 trig ratios (a) 2 sin5x cos3x (b) 2 cos4x cosx (c) sin4x cos5x (d) sin2x sinx (4) Solve the following equations, for values of x between 0 and 360 o (a) 2sinx + sin2x =0 (b) sinx + sin3x + sin5x = 0 (c) 2cos2x + 2sin2x – 1 = 0 (d) tan2x - 5tanx = 0 (5) Solve the following equations, for values of x between 0 and 360 o (a) 3sinx + 4cosx = 1 (b) sinx + cosx = 0.5 (c) 3sinx - 5cosx = 5 (d) 3sinx - sinx = 1 (e) 3sin2x +4cos2x = 1 (6) If sinx = 0.8 and cos x =-0.6, determine the value of the following (a) sin2x (b) cos4x (c) sin(x + 45) (d) sin(x +120) + sin(x -120)