Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help
cos (4pi/3 – 3pi/4)
To evaluate the expression cos (4π/3 – 3π/4) and find the exact value, we can use the cosine difference identity:
cos (a - b) = cos a * cos b + sin a * sin b
Let's apply this identity to the given expression:
cos (4π/3 – 3π/4) = cos (4π/3) * cos (3π/4) + sin (4π/3) * sin (3π/4)
We know the exact values for cos (π/3), cos (π/4), sin (π/3), and sin (π/4). So, we'll use these values:
cos (π/3) = 1/2
cos (π/4) = √2/2
sin (π/3) = √3/2
sin (π/4) = √2/2
Now, substitute the values into our expression:
cos (4π/3 – 3π/4) = (1/2) * (√2/2) + (√3/2) * (√2/2)
Multiply the fractions:
cos (4π/3 – 3π/4) = (√2/4) + (√6/4)
Adding the fractions:
cos (4π/3 – 3π/4) = (√2 + √6) / 4
Therefore, the exact value of cos (4π/3 – 3π/4) is (√2 + √6) / 4.