use the sum and difference identities to find the cosine angle
cos pi/9 cos pi/3 - sin pi/9 sin pi/3
I do not know how to solve this because pi/9 is not on the unit circle.
but it gave you a hint, ...
cos(A+B) = cosAcosB - sinAsinB
does the right side not follow the pattern of your question?
so ...
cos pi/9 cos pi/3 - sin pi/9 sin pi/3
= cos(pi/9 + pi/3)
= cos(4pi/9)
You are right, 4pi/9 radians, or 80 degrees, is not one of the standard angles on the unit circle, nor is it obtainable using combinations of our standard angles.
To find the value of the expression cos(pi/9) cos(pi/3) - sin(pi/9) sin(pi/3), we can use the sum and difference identities of trigonometric functions.
The cosine of the sum or difference of two angles, A and B, is given by the formula:
cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B)
In this case, we can rewrite the given expression as:
cos(pi/9) cos(pi/3) - sin(pi/9) sin(pi/3)
Using the sum and difference identity for cosine, we can rewrite pi/3 as pi/6 + pi/6:
cos(pi/9) [cos(pi/6 + pi/6) - sin(pi/9) sin(pi/3)
Applying the sum identity for cosine:
cos(pi/9) [cos(pi/6)cos(pi/6) - sin(pi/6)sin(pi/6)] - sin(pi/9) sin(pi/3)
Since cos(pi/6) = √3/2 and sin(pi/6) = 1/2:
cos(pi/9) [(√3/2)(√3/2) - (1/2)(1/2)] - sin(pi/9) sin(pi/3)
Simplifying, we have:
cos(pi/9) [(3/4) - (1/4)] - sin(pi/9) sin(pi/3)
cos(pi/9) (2/4) - sin(pi/9) sin(pi/3)
This can be further simplified to:
(1/2)cos(pi/9) - sin(pi/9) sin(pi/3)
Now, we can evaluate this expression using a calculator or trigonometric tables. Plug in the value of pi/9 and pi/3 to find the final result.