Solving for Cos(A - B) Using Trigonometric Identities and Values

Given the trigonometric equations cos(AB) frac{1}{2} and sin A frac{1}{sqrt{2}}, we aim to find the value of cos(A - B). This process involves several steps using trigonometric identities, particularly the cosine of sum and difference identities, and the Pythagorean identity.

Step 1: Applying the Pythagorean Identity

First, we use the Pythagorean identity to find cos A

The Pythagorean identity states:

sin^2 A cos^2 A 1

Given that sin A frac{1}{sqrt{2}}, we can find cos A

(sin^2 A cos^2 A 1)

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

(frac{1}{2} cos^2 A 1)

(cos^2 A frac{1}{2})

(cos A pm frac{1}{sqrt{2}})

Step 2: Identifying Possible Angles for A - B

Given that cos(AB) frac{1}{2}, we know that the possible angles for A - B are:

A - B 60^circ 360^circ k or A - B 300^circ 360^circ k, where k in mathbb{Z}

For simplicity, we will consider the case where:

A - B 60^circ

Step 3: Solving for B

Given A - B 60^circ, we can express B in terms of A:

B 60^circ - A

Step 4: Using the Cosine and Sine Subtraction Formulas

Using the cosine and sine subtraction formulas:

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

We need to find cos B) and (sin B

cos B cos(60^circ - A) cos 60^circ cos A sin 60^circ sin A frac{1}{2} cos A frac{sqrt{3}}{2} sin A

sin B sin(60^circ - A) sin 60^circ cos A - cos 60^circ sin A frac{sqrt{3}}{2} cos A - frac{1}{2} sin A

Step 5: Substituting the Values of (cos A) and (sin A)

Given that cos A frac{1}{sqrt{2}} (or -frac{1}{sqrt{2}}), we substitute these values into the equations for (cos B) and (sin B:

cos B frac{1}{2} cdot frac{1}{sqrt{2}} frac{sqrt{3}}{2} cdot frac{1}{sqrt{2}} frac{1}{2sqrt{2}} frac{sqrt{3}}{2sqrt{2}} frac{1 sqrt{3}}{2sqrt{2}}

sin B frac{sqrt{3}}{2} cdot frac{1}{sqrt{2}} - frac{1}{2} cdot frac{1}{sqrt{2}} frac{sqrt{3}}{2sqrt{2}} - frac{1}{2sqrt{2}} frac{sqrt{3} - 1}{2sqrt{2}}

Step 6: Calculating (cos(A - B))

Now, substituting the values of cos B and sin B into the equation for cos(A - B)

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

frac{1}{sqrt{2}} cdot frac{1 sqrt{3}}{2sqrt{2}} frac{1}{sqrt{2}} cdot frac{sqrt{3} - 1}{2sqrt{2}}

frac{1 sqrt{3}}{2cdot 2} frac{sqrt{3} - 1}{2cdot 2} frac{1 sqrt{3} sqrt{3} - 1}{4} frac{2sqrt{3}}{4} frac{sqrt{3}}{2}

Conclusion

Thus, the value of cos(A - B) is found to be:

cos(A - B) frac{sqrt{3}}{2}