Is it possible to find the value of a trigonometric function without finding the actual angle?
Example: sin A = ?
It is possible! But provided some other information is given.
If the question is :
Find sin A without finding the angle A, given that cos A = 4 / 5.
Here, you can get the answer through the use of Pythagorus' Theorem.
You need to know the other length of the unknown side (x) of the right-angled triangle to get the final answer of sin A.
cos A = 4 / 5 implies x2 + 42 = 52
x = sqrt(25 - 16) = 3
Therefore knowing now that x (the other unknown side) is 3, sin A = 3 / 5 (the final answer).
This is a useful technique to compute the trigonometric value without finding the angle.
We can expand this technique to include more complex question. See the below example.
Find cos (A + B), without finding the value of angles A & B, given cos A = 4 / 5 and sin B = 3 / 13.
Again making use of the powerful Pythagorus' Theorem, we can find that
sin A = 3 / 5 and cos B = 12 / 13.
cos ( A + B ) = cos A cos B - sin A sin B
= (4/5)(12/13) - (3/5)(3/13) = 39 / 65 (Answer)
Note, this was done without finding the actual values of the angles A and B !
Learn this trick and you can save yourself some valuable time.
That's the power of knowing maths concept and its principles. The little tricks come from there!