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A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
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Solved Examples on Trigonometric Equations and...