INTRODUCTION:

Trigonometry is the relation between the sides and angle of a triangle. By default, we consider the triangle to be a right-angled triangle. Generally, there are 6 trigonometric functions, which are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (cosec).

FUNCTIONS

Some important formulae:

Domain and Range of the functions:

The angle can be represented in either degrees(°) or radians(rad). The relation between radian and degree is shown as:

SUM AND DIFFERENCE OF ANGLES

Let A and B are 2 angles.

Sin(A+B) = sinAcosB+cosAsinB

Sin(A-B) = sinAcosB-cosAsinB

Cos(A+B) = cosAcosB-sinAsinB

Cos(A-B) = cosAcosB+sinAsinB

Tan(A+B) = (tanA+tanB)/(1-tanAtanB)

Tan(A-B) = (tanA-tanB)/(1+tanAtanB)

DOUBLE ANGLES

Substitute B as A in the above equations,

Sin 2A= 2sinAcosA

Cos2A= cos²A-sin²A= 1-2sin²A= 2cos²A-1

Tan2A= 2tanA/(1-tan²A)

HALF-ANGLES

If we consider 2A=x, then A=x/2.

Sin x= 2 sin(x/2)cos(x/2)

Cos x= 2cos²(x/2)-1, that is, cos²(x/2)=1-cosx

Cos x=1- 2sin²(x/2), that is, sin²(x/2)= 1+cosx

Standard values known to us

From the standard values and the formulae, we can easily find out certain angles like 15°,22.5° and so on

SUM OF 2 SAME TRIGONOMETRIC IDENTITIES

SinC+ sinD= 2 sin{(C+D)/2}cos{(C-D)/2}

SinC-SinD= 2sin{(C-D)/2}cos{(C+D)/2}

CosC+cosD= 2cos{(C+D)/2}cos{(C-D)/2}

CosC-cosD= 2sin{(C+D)/2}sin{(D-C)/2}

Some Fundamental Identities

Sin(-x)= -sin(x)

Cos(-x)=cos(x)

Tan(-x)= -tan(x)

General solutions of trigonometric equations

Our primary aim will be to convert the equation into same trigonometric identity(like terms containing cos 2x and sin 2x can be converted to functions of cosx). Then, we will solve the equation using our trigonometric general solutions. While solving the equations, we should keep in mind about the ranges of the trigonometric functions discussed above.