# Trigonometric Equations

## Trigonometric Equations Synopsis

**Synopsis**

- If a point on a unit circle is on the terminal side of an angle in the standard position, then the sine of such an angle is simply the y-coordinate of the point and the cosine of the angle is the x-coordinate of that point.
- All the angles which are integral multiples of are called quadrantal angles. Values of quadrantal angles are:

**Trigonometric functions as even or odd**

• A function f(x) is said to be an even function if f(-x) = f(x), for all x in its domain.

• A function f(x) is said to be an odd function if f(-x) = -f(x), for all x in its domain.

• Cosine is an even function and sine is an odd function since cos(−x) = cos x and sin(−x) = −sin x**Sign of trigonometric functions in various quadrants**

• In quadrant I, all the trigonometric functions are positive.

• In quadrant II, only sine and cosec functions are positive.

• In quadrant III, only tan and cot functions are positive.

• In quadrant IV, only cosine and sec functions are positive.

• This is depicted as follows:

• In quadrants, where the y-axis is positive (i.e. I and II), sine is positive, and in quadrants where the x-axis is positive (i.e. I and IV), cosine is positive.

• A simple rule to remember the sign of the trigonometrical ratios, in all the four quadrants, is the four letter phrase—All Silver Tea Cups.**Period of a function**A function ‘f’ is said to be a periodic function if there exists a real number T > 0 such that f(x + T) = f(x) for all ‘x’. This ‘T’ is the period of function.**Trigonometric ratios of complementary angles**i. sin (90° - θ) = cos θ

ii. cos (90° - θ) = sin θ

iii. tan (90° - θ) = cot θ

iv. cosec (90° - θ) = sec θ

v. sec (90° - θ) = cosec θ

vi. cot (90° - θ) = tan θ**Trigonometric ratios of (90° + θ) in terms of θ**i. sin (90° + θ) = cos θ

ii. cos (90° + θ) = -sin θ

iii. tan (90° + θ) = -cot θ

iv. cosec (90° + θ) = sec θ

v. sec (90° + θ) = -cosec θ

vi. cot (90° + θ) = -tan θ**Trigonometric ratios of (180° - θ) in terms of θ**i. sin (180° - θ) = sin θ

ii. cos (180° - θ) = -**cos**θ

iii. tan (180° - θ) = -**tan**θ

iv. cosec (180° - θ) = cosec θ

v. sec (180° - θ) = -**sec**θ

vi. cot (180° - θ) = -cot θ**Trigonometric ratios of (180° + θ ) in terms of θ**i. sin (180° + θ) = -sin θ

ii. cos (180° + θ) = -cos θ

iii. tan (180° + θ) = tan θ

iv. cosec (180° + θ) = -cosec θ

v. sec (180° + θ) = -sec θ

vi. cot (180° + θ) = cot θ**Trigonometric ratios of (360° - θ ) in terms of θ**

i. sin (360° - θ) = -sin θ

ii. cos (360° - θ) =**cos**θ

iii. tan (360° - θ) = -**tan**θ

iv. cosec (360° - θ) = -cosec θ

v. sec (360° - θ) =**sec**θ

vi. cot (360° - θ) = -cot θ**Trigonometric ratios of (360° + θ ) in terms of θ**i. sin (360° + θ) = sin θ

ii. cos (360° + θ) = cos θ

iii. tan (360° + θ) = tan θ

iv. cosec (360° + θ) = cosec θ

v. sec (360° + θ) = sec θ

vi. cot (360° + θ) = cot θ**General and particular solutions**• The solutions of a trigonometric equation, for which 0 ≤ x ≤ 2, are called**principal solutions**.

• The expression involving the integer ‘n’, which gives all the solutions of a trigonometric equation, is called the**general solution**.**Principal value**• The numerically smallest value of the angle (in degree or radian) satisfying a given trigonometric equation is called the**Principal Value**.

• If there are two values, one positive and the other negative, which are numerically equal, then the positive value is taken as the Principal Value.**Trigonometric equations****Sine Rule:**The sine rule states that**Laws of Cosine****Domain and range of various trigonometric functions****Graphs of trigonometric functions**Graphs help in the visualization of the properties of trigonometric functions.

The graph of y = sinθ can be drawn by plotting a number of points (θ, sinθ) as θ takes a series of different values.

i. Graph of sin x

ii. Graph of cos x

iii. Graph of tan x

iv. Graph of sec x

v. Graph of cosec x

vi. Graph of cot x

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