Solving for Cosine of Angle Sum Given Trigonometric Values

Given the trigonometric values tan A 1 and sin B frac{1}{sqrt{2}}, find the value of cos (A B). This problem involves understanding fundamental trigonometric concepts and applying the cosine addition formula. Letrsquo;s break down the steps to solve this problem.

Step 1: Identify Angle Values

For tan A 1:
- This implies that A 45^circ or A frac{pi}{4} radians, since tan 45^circ 1.

For sin B frac{1}{sqrt{2}} frac{sqrt{2}}{2}:
- This implies that B 45^circ or B 135^circ, since sin 45^circ frac{1}{sqrt{2}}.

Step 2: Apply the Cosine Addition Formula

The cosine addition formula is given by:

cos (A B) cos A cos B - sin A sin B

Step 3: Determine cos A, sin A, cos B, and sin B

Since A 45^circ:

cos A cos 45^circ frac{1}{sqrt{2}} sin A sin 45^circ frac{1}{sqrt{2}}

Since B 45^circ or B 135^circ:

cos B frac{1}{sqrt{2}} sin B frac{1}{sqrt{2}}
or cos B -frac{1}{sqrt{2}} sin B frac{1}{sqrt{2}}

Step 4: Substitute Values into the Formula

Now, we substitute these values into the cosine addition formula:

cos (A B) left(frac{1}{sqrt{2}}right) left(frac{1}{sqrt{2}}right) - left(frac{1}{sqrt{2}}right) left(frac{1}{sqrt{2}}right)

cos (A B) frac{1}{2} - frac{1}{2} 0

Thus, the value of cos (A B) is 0.

Alternate Solutions

Letrsquo;s also consider other possible values for angles A and B based on the given trigonometric values:

When A 225^circ and B 45^circ: cos (A B) cos 270^circ 0 When A 45^circ and B 135^circ: cos (A B) cos 180^circ -1 When A 225^circ and B 135^circ: cos (A B) cos 360^circ 1

These results show the importance of considering all possible values of angles in trigonometric problems.

Summary

To find the value of cos (A B), we utilized the trigonometric values tan A 1 and sin B frac{1}{sqrt{2}}. By identifying the appropriate angles and applying the cosine addition formula, we found that cos (A B) can be 0, -1, or 1. Understanding the properties of trigonometric functions and the cosine addition formula are crucial for solving such problems.