Solving Trigonometric Functions

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles.

The most familiar trigonometric functions are the sine, cosine, and tangent. Trigonometric functions can also be called as circular functions.

A few example problems are given below to learn solving trigonometric functions.

(Source: Wikipedia)

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Example problems for solving trigonometric functions:

Example 1:

Prove that  (cos6x + cos 4x)/(sin 6x - sin4x) = cot x

Solution:

To prove the above equation,  take the left hand side (LHS) of the equation and simplify it step by step to get right hand side of the equation (RHS).

Step 1: Given

(cos6x + cos 4x)/(sin 6x - sin4x) = cot x

Step 2: Take left hand side of the equation and prove it

LHS = (cos6x + cos 4x)/(sin 6x - sin4x)

= (2cos((6x+4x)/2)cos((6x-4x)/2))/(2cos((6x+4x)/2)sin((6x-4x)/2))       [Using trigonometric formulas]

Solving above function, we get

= (2cos5xcosx)/(2cos5xsinx)

=  cosx/sinx

= cot x

= RHS

Hence proved

Example 2:

Prove that  (sin3x - sinx)/(cosx - cos3x) = cot 2x

Solution:

To prove the above equation,  take the left hand side (LHS) of the equation and simplify it step by step to get right hand side of the equation (RHS).

Step 1: Given

 (sin3x - sinx)/(cosx - cos3x)  = cot 2x

Step 2: Take left hand side of the equation and prove it

LHS = (sin3x - sinx)/(cosx - cos3x)

= (2cos((3x+x)/2)sin((3x-x)/2))/(-2sin((x+3x)/2)sin((x-3x)/2))         [Using trigonometric formulas]

Solving above function, we get

= (cos2xsinx)/(-sin2xsin(-x))

= (cos2xsinx)/(sin2xsinx)

= cot 2x

= RHS

Hence proved

Example 3:

Solve the function (sin3x-sinx)/(cos2x)

Solution:

Given:

(sin3x-sinx)/(cos2x)

Solving above function using trigonometric formula sin C - sin D = 2cos(C+D)/2 sin(C - D)/2 , we get

(sin3x-sinx)/(cos2x) = (2cos((3x+x)/2)sin((3x-x)/2))/(cos2x)

= (2cos2xsinx)/(cos2x)

= 2sin x

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Example 4:

Solve (sinx - siny)/(cosx + cosy)

Solution:

We can write (sinx - siny)/(cosx + cosy) as follows using trigonometric identities,

(sinx - siny)/(cosx + cosy) = (2cos((x+y)/2)sin((x-y)/2))/(2cos((x+y)/2)cos((x-y)/2))

= tan((x-y)/2)

Practice problems for solving trigonometric functions:

1) Prove that sin x + sin 3x + sin 5x + sin 7x = 4sin 4x cos 2x cos x

2) Prove that (sin5x - 2sin3x + sinx)/(cos5x - cosx) = cosec 2x - cot 2x

3) Simplify (sinx + sin3x)/(cosx - cos 3x)

Ans: cot x