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Notes On Trigonometric Equations And Identities:
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 โTโ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2ฯ
2) cos x has period 2ฯ
3) tan x has period ฯ
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2ฯ/a
5) tan (ax+b) and cot (ax+b) have ฯ/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period ฯ/a
7) |tan (ax+b)| and |cot (ax+b)| have ฯ/2a as their period
โSum and Difference Formulae of Trigonometric Ratios
1) sin(a + ร) = sin(a)cos(ร) + cos(a)sin(ร)
2) sin(a โ ร) = sin(a)cos(ร) โ cos(a)sin(ร)
3) cos(a + ร) = cos(a)cos(ร) โ sin(a)sin(ร)
4) cos(a โ ร) = cos(a)cos(ร) + sin(a)sin(ร)
5) tan(a + ร) = [tan(a) + tan (ร)]/ [1 - tan(a)tan (ร)]
6)tan(a - ร) = [tan(a) - tan (ร)]/ [1 + tan (a) tan (ร)]
7) tan (ฯ/4 + ฮธ) = (1 + tan ฮธ)/(1 - tan ฮธ)
8) tan (ฯ/4 - ฮธ) = (1 - tan ฮธ)/(1 + tan ฮธ)
9) cot (a + ร) = [cot(a) . cot (ร) - 1]/ [cot (a) +cot (ร)]
10) cot (a - ร) = [cot(a) . cot (ร) + 1]/ [cot (ร) - cot (a)]
โDouble or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x โ sin2x = 1 โ 2sin2x = 2cos2x โ 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x โ 4 sin3x
5) cos3x = 4 cos3x โ 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
โFor angles A, B and C, we have
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 โtan A tan B tan C]/ [1- tan Atan B - tan B tan C โtan A tan C
4) cot (A + B +C) = [cot A cot B cot C โ cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotCโ1]
โList of some other trigonometric formulas:
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 ยฑ tanB = sin (A ยฑ B)/ cos A cos B
10)cot A ยฑ cot B = sin (B ยฑ A)/ sin A sin B
โMethod of solving a trigonometric equation:
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.
โSome results which are useful for solving trigonometric equations:
1) sin ฮธ = sina and cosฮธ = cosa โ ฮธ = 2nฯ + a
2) sin ฮธ = 0 โ ฮธ = nฯ
3) cosฮธ = 0 โ ฮธ = (2n + 1)ฯ/2
4) tan ฮธ = 0 โ ฮธ = nฯ
5) sinฮธ = sinaโ ฮธ = nฯ + (-1)na where a โ [โฯ/2, ฯ/2]
6) cosฮธ= cos a โ ฮธ = 2nฯ ยฑ a, where a โ[0,ฯ]
7) tanฮธ = tanaโ ฮธ = nฯ+ a, where a โ[โฯ/2, ฯ/2]
8) sinฮธ = 1 โ ฮธ= (4n + 1)ฯ/2
9) sin ฮธ = -1 โ ฮธ = (4n - 1) ฯ /2
10) sin ฮธ = -1 โ ฮธ = (2n +1) ฯ /2
11) |sinฮธ| = 1โ ฮธ =2nฯ
12) cosฮธ = 1 โ ฮธ =(2n + 1)
13) |cosฮธ| = 1โ ฮธ =nฯ
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