Quandaries and Queries


Grade 11

How do you find the exact solution to cosine 35 degrees.

Hi Jason,

The exact value of cos (35 deg) cannot be written down "exactly," although you can compute it to as many decimal places as you wish. Mathematicians working 2000 years ago worked out the value using trigonometric identities. Your calculator uses a similar method, but it takes advantage of ideas such as "Taylor series" that come from calculus. As a simplified version of their method, you could compute the sine and cosine of 5 degrees, then use the angle-sum formula for cos (30 + 5).

For sin 5 you plug x = 5*pi/180 into 1 - x2/2x4/24.

For cos 5 you plug x = 5*pi/180 into x - x3/6.

Since cos 30 = sqrt(3)/2 and sin(30) = 1/2, you can compute cos 35 as

cos 35 = sqrt(3)/2*cos 5 - 1/2*sin 5.


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