how to determine exact value?

Question

determine the exact value of cos (5π/12)

 

1.) √2/4 + √3/4

 

2.) √3 -1/ 4

 

3.) √6/4 - √2/4

 

4.) 1/2 + √3/4

 

5.) √6/4 + √2/4

 

6.) √2+√3/2

Answer 1

cos(5pi/12)

cos(900/12)

cos(75)

cos(75)=cos(30+45)

cos(30+45)=cos(30)cos(45)-sin(30)sin(45)

                 =(√3/2)(1/√2)-(1/2)(1/√2)

                 =(√3/2√2)-(1/2√2)

                 =(√3-1)/(2√2)

                 =(2√2)(√3-1)/(2√2)(2√2)

                 =(2√6-2√2)/8

                 =2√6/8-2√2/8

                 =√6/4-√2/4

the answer is 3.)

Answer 2

starting with the following formula:

 

cos(∏/2-θ) = sin(θ)

 

we can simplify the original equation to:

 

cos(5Π/12) = cos (6Π/12 - Π/12) = sin(Π/12)

 

we know that cos(Π/6) = √3/2 and also:

 

sin(θ) = √((1-cos(2θ))/2)

 

substituting θ by Π/12 gives us:

 

sin(Π/12) = √((1-cos(Π/6))/2) = √((1-√3/2 )/2) = √((2-√3)/4) = √(2-√3)/2

 

so we can say:

 

cos(5Π/12) = √(2-√3)/2

 

if it was asking for the cos²(5Π/12) then the answer would have been 1/2-√3/4