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)
0
