## Thursday, 14 August 2008

### Finding Trigonometric Value Without Evaluating The Angle

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.

Why?

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.

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!
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