How can the value of cos 15 be found without the use of a calculator. Which formula has to be used?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The value of cos 30 is usually known to all students.

cos 30 = (sqrt 3)/2.

Now we use the formula for cos 2x, which is:

cos 2x = (cos x)^2 - (sin x)^2

=> cos 2x =  2 (cos x)^2 - 1

Now substitute 15 for x here.

We have cos (30) = 2 (cos 15)^2 - 1

=>  (sqrt 3)/2 = 2 (cos 15)^2 - 1

=> 1 + (sqrt 3)/2 = 2 (cos 15)^2

=> (1 + (sqrt 3)/2)/2 = (cos 15)^2

=> cos 15 = sqrt [ (1 + (sqrt 3)/2)/2]

=> cos 15 = sqrt [ 1/2 + (sqrt 3)/4]

You will most probably need a calculator to find the value of the square root here, but you won't need one which has values of cosine stored in it.

Therefore cos 15 can be calculated without a calculator as sqrt [ 1/2 + (sqrt 3)/4]

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial