Question

Use the half-angle identities to find the exact value of cos 15 degrees.

This is what I have so far: 

cos15 degrees = cos1/2(30 degree) = sqrt (1+cos30)/2 = sqrt (1+ sqrt3/2)/ 2

But.. I don't understand how the cos30 turns into sqrt 3/2??

Answer 1

The question is: how does cos(30) = $\frac{ \sqrt{3} }{2}$


Let's start by considering the equilateral triangle ABC in the diagram below. An equilateral triangle has equal size of angles, 60°, 60°, and 60°. Let's say the length of each side to be 2 units.

If we split this triangle into two congruent right-angled triangle, we will have the measurement of the perpendicular height to be √3 by Pythagoras theorem.

By trigonometry ratio we then have $cos(30) = \frac{ \sqrt{3} }{2}$

This ratio will always work for any right angle triangle with angles 90°, 60° and 30°

Going back to the original question:

cos(15°) = cos(30°) which is cos($\frac{30}{2}$)
let θ be 30° then cos(15°)=cos(θ/2)
by the trigonometry formula for half angle identity
cos(θ/2) = √(1+cosθ)/2
cos(15) = √(1+cos(30))/2
cos(15) = √(1+$\frac{ \sqrt{3}} {2}$) / 2
cos(15) = $\frac{ \sqrt{2+ \sqrt{3} } }{2}$

Answer 2

The cosine value of cos(15) is $\sqrt$(1 + $\sqrt$3/2)/2

The trigonometry identity of half angles is given as:

cos($\theta$/2) = $\sqrt{$(1 + cos($\theta$))/2

Substitute 30 for $\theta$

So, the equation becomes

cos(30/2) = $\sqrt$(1 + cos(30))/2

In trigonometry, we have:

cos(30) = $\sqrt$3/2

So, we have:

cos(30/2) = $\sqrt$(1 + $\sqrt$3/2)/2

Divide 30 by 2

cos(15) = $\sqrt$(1 + $\sqrt$3/2)/2

Hence, the cosine value of cos(15) is $\sqrt$(1 + $\sqrt$3/2)/2

Read more about trigonometry identities at:

https://brainly.com/question/7331447