Solving Examples of Trigonometry

Trigonometry is one of the branch in mathematics, it is having so many examples and applications. Trigonometry is mainly used in civil engineering,it is very useful in engineering field. it is having angles, using the angles we can  use in construction field.Its applications is used in so many math divisions.

 Here we are discussing the examples of trigonometry 

Basic formulas in trigonometry:

 

Solved examples of trigonometry

 

Example1:

 Show that cos 3A = 4cos^3A - 3cosA 

Proof :

cos(3A) = cos(2A+A)

We know formula cos(A+B) = cosAcosB - sinAsinB
                    Cos ( 2A + A ) = cos(2A)cos(A) - sin(2A)sinA

 cos(2A) = cos(A+A)

= cos²A - sin²A

= cos²A - (1-cos²A)

= 2cos²A - 1

Hence, Cos 3A = (2cos²A-1)cosA - 2sin²AcosA

                    = 2cos³A - cosA - 2(1-cos²A)cosA

                            = 2cos³A - cosA - 2cosA + 2cos³A

                      = 4cos³A - 3cosA

So LHS= RHS

 

Example 2:

Find the exact value of 1. tan 30  2. cos 60 and 3. cos 45

Solution:

Here we using right angle triangle properties

A 30-60-90 right triangle has sides that are in the ratio of

1 : √3 : 2

And, a 45-45-90 right triangle has sides that are

1 : 1 : √2

From these ratios

tan 30 º = (√3)/3

cos 60 º = 1/2

cos 45 º = (√2)/2

Example3:

 Prove that cos(60+A)cos(60-A)-sin(60+A)sin(60-A)= 0.5

Solution:

cos(60+A)cos(60-A)-sin(60+A)sin(60-A) 

= cos(60+A+60-A)

= cos(120)

=Cos(180-60 )

= cos60

= 1/2 

  = 0.5

So LHS= RHS

Example 4:

What is the exact value of cos150

Solution:

cos 150 = cos (180 - 30)

We know formula cos (A - B) = cos A cos B + sin A sin B

cos 150 = cos 180 cos 30 + sin 180 sin 30

cos 150 = (-1) cos 30 + (0) sin 30

cos 150 = - cos 30

                = -`sqrt(3)/2`