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Trigonometric Functions Class 11 Notes Maths Chapter 3
- 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 - Sum and Difference Formulae of Trigonometric Ratios
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)] - Double or Triple -Angle Identities
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) - For 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 –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] - List 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 ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B - Method 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. - Some results which are useful for solving trigonometric equations:
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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FAQs on Trigonometric Functions Class 11 Notes Maths Chapter 3
| 1. What are trigonometric equations? |  |
| 2. How do you solve trigonometric equations? | |
Ans. To solve trigonometric equations, we use various trigonometric identities and properties. These equations are solved by simplifying them using these identities and then finding the values of the angles that satisfy the equation within a given range.
| 3. What are trigonometric identities? | |
Ans. Trigonometric identities are equations that are true for all values of the variables involved. They are used to simplify trigonometric expressions and equations. Some common identities include Pythagorean identities, reciprocal identities, quotient identities, and co-function identities.
| 4. How do you prove trigonometric identities? | |
Ans. Trigonometric identities can be proven using algebraic manipulations and the properties of trigonometric functions. The most common method is to start with one side of the equation and manipulate it until it is equivalent to the other side of the equation, using the known trigonometric identities.
| 5. What is the importance of trigonometric equations and identities in JEE exams? | |
Ans. Trigonometric equations and identities are important in JEE exams as they are frequently tested in the mathematics section. Understanding and applying these equations and identities is crucial for solving trigonometry problems and scoring well in the exam.
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