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```Revision Notes on Trigonometric Functions and Equations

The trigonometric ratios are the ratios between the two sides of a right-angled triangle with respect to an angle and hence they are real numbers.

The angle θ taken into consideration may be acute, obtuse or right angle.

Various trigonometry formulas related to the basic trigonometric ratios which must be remembered include:

1) sin2 θ + cos2 θ = 1

2) sec2 θ – tan2 θ = 1

3) cosec2 θ - cot2 θ = 1

4) sin θ cosec θ = tan θ cot θ = sec θ cos θ = 1

5) sin2 θ + cos2 θ = 1, so each of them is numerically less than 1.

6) |sin θ| ≤ 1 and |cos θ| ≤ 1

7) -1 < cosec θ < 1 and -1 < sec θ < 1

8) tan θ and cot θ may take any value.

The trigonometry ratios can be either positive or negative depending on the quadrant in which they lie. The figure given below illustrates the various trigonometric ratios and their signs in various quadrants:

Range of f : X →Y is the set of all images f(x) which belong to Y, i.e.

Range f = {f(x) ∈ Y: x ∈ X} ⊆ Y.

The domain and range of various trigonometric ratios are given below:

Trigonometric Function

Domain

Range

sin x

R

-1 ≤ sin x ≤ 1

cos x

R

-1 ≤ cos x ≤ 1

tan x

R – {(2n + 1)π/2, n ∈ I}

R

cosec x

R – {nπ, n ∈ I}

R – {x: -1 < x < 1}

sec x

R – {(2n + 1)π/2, n ∈ I}

R – {x: -1 < x < 1}

cot x

R – {nπ, n ∈ I}

R

Given below is the trigonometry table of the basic ratios for some frequently used angles. These formulas must be memorized as they are used in almost all questions:

Angle (x)

sin x

cos x

tan x

cosec x

sec x

cot x

0°

0

1

0

undefined

1

undefined

90° = π/2

1

0

undefined

1

undefined

0

180° = π

0

-1

0

undefined

-1

undefined

270° =3π/2

-1

0

undefined

-1

undefined

0

360° = 2π

0

1

0

undefined

1

undefined

Some other formulas of trigonometry which are useful in solving questions:

Angles

0°

30°

45°

60°

90°

sin

0

1/2

1/√2

√3/2

1

cos

1

√3/2

1/√2

1/2

0

tan

0

√3/2

1

√3

undefined

cosec

undefined

2

√2

2/√3

1

sec

1

2/√3

√2

2

undefined

cot

undefined

√3

1

1/√3

0

The values of various trigonometric ratios vary in different quadrants. The variation is listed below quadrant wise:

sine - increases from 0 to 1

cosine - decreases from 1 to 0

tangent - increases from 0 to ∞

cotangent - decreases from ∞ to 0

secant - increases from 1 to ∞

cosecant - decreases from ∞ to 1

sine - decreases from 1 to 0

cosine - decreases from 0 to -1

tangent - increases from -∞ to 0

cotangent - decreases from 0 to -∞

secant - increases from -∞ to -1

cosecant - increases from 1 to ∞

sine - decreases from 0 to -1

cosine - increases from -1 to 0

tangent - increases from 0 to ∞

cotangent - decreases from ∞ to 0

secant - decreases from -1 to -∞

cosecant - increases from -∞ to -1

sine - increases from -1 to 0

cosine - increases from 0 to 1

tangent - increases from -∞ to 0

cotangent - decreases from 0 to -∞

secant - decreases from ∞ to 1

cosecant - decreases from -1 to ∞

The reciprocal functions have the same period as that of the original functions.

If the period of f(x) is T then that of k f(ax+b) is T/mod (a), hence period is affected by coefficient of x only.

If f(x) has its period T and g(x) has its period M, then (af(x) + bg(x)) has its period < L.C.M. (T, M). Moreover if f(x) and g(x) are basic trigonometric functions then period of [af (x) + bg(x)] = L.C.M. (T, M).

If a constant is added, subtracted, multiplied or divided in a periodic function, its period is unaffected.

Graphs of some basic trigonometric functions:

1) sin x

2) cos x

3) tan x

4) sec x

5) cosec x

6) cot x

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 ‘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) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C - sin A sin B sin C

2) cos (A + B + C) = cos A cos B cos C - cos A sin B sin C - sin A cos B sin C - sin A sin B cos C

3) tan (A + B + C) = [tan A + tan B + tan C – tan A tan B tan C]/ [1 - tan A tan B - tan B tan C – tan A tan C

4) cot (A + B + C) = [cot A cot B cot C – cot A - cot B - cot C]/ [cot A cot B + cot B cot C +  cot A cot C – 1]

List of some other trigonometric formulas:

1) 2sin A cos B = sin (A + B) + sin (A - B)

2) 2cos Asin B = sin (A + B) - sin (A - B)

3) 2cos Acos B = cos (A + B) + cos (A - B)

4) 2sin Asin B = 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) cos A + cos  B = 2 cos [(A + B)/2] cos [(A - B)/2]

8) cos A - cos B = 2 sin [(A + B)/2] sin [(B – A)/2]

9) tan A ± tan B = 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 = sin a and cos θ = cos a ⇒ θ = 2nπ + a

2) sin θ = 0 ⇒ θ = nπ

3) cosθ = 0 ⇒ θ = (2n + 1)π/2

4) tan θ = 0 ⇒ θ = nπ

5) sin θ = sin a ⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]

6) cosθ = cos a ⇒ θ = 2nπ ± a, where a ∈ [0,π]

7) tan θ = tan a ⇒ θ = nπ + a, where a ∈ [–π/2, π/2]

8) sin θ = 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) |sin θ| = 1⇒ θ =2nπ

12) cos θ = 1 ⇒ θ = (2n + 1)

13) |cos θ| = 1 ⇒ θ = nπ
```
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