Let's solve this problem involving trigonometric equations and finding the measures of acute angles A and B.

Understanding the Problem: Finding Angles from Cosine Values

We are given two equations involving the cosine of the sum and difference of two positive acute angles A and B, with the condition that A is greater than or equal to B. The equations are:

• \( \cos (A - B) = \frac{{\sqrt 3 }}{2} \)
• \( \cos (A + B) = 0 \)

Our goal is to find the specific degree measures of angles A and B that satisfy these conditions.

Step-by-Step Solution

Step 1: Determine the values of (A - B) and (A + B)

We use the given cosine values to find the angles whose cosine is equal to those values.

For the first equation, \( \cos (A - B) = \frac{{\sqrt 3 }}{2} \). We know that \( \cos 30^\circ = \frac{{\sqrt 3 }}{2} \). Since A and B are positive acute angles and \(A \ge B\), the angle \(A - B\) must be in the range \(0^\circ \le A-B < 90^\circ\). Therefore, the principal value is appropriate.

So, we get our first linear equation:

\[ A - B = 30^\circ \quad \text{(Equation 1)} \]

For the second equation, \( \cos (A + B) = 0 \). We know that \( \cos 90^\circ = 0 \). Since A and B are positive acute angles, their sum \(A + B\) must be in the range \(0^\circ < A+B < 180^\circ\). The value of \(A+B\) in this range for which \( \cos (A + B) = 0 \) is \(90^\circ\).

So, we get our second linear equation:

\[ A + B = 90^\circ \quad \text{(Equation 2)} \]

Step 2: Solve the system of linear equations for A and B

Now we have a system of two linear equations with two variables, A and B:

1. \( A - B = 30^\circ \)

2. \( A + B = 90^\circ \)

We can solve this system using the elimination method by adding Equation 1 and Equation 2:

\[ (A - B) + (A + B) = 30^\circ + 90^\circ \]

\[ A - B + A + B = 120^\circ \]

\[ 2A = 120^\circ \]

Divide by 2 to find A:

\[ A = \frac{120^\circ}{2} \]

\[ A = 60^\circ \]

Now substitute the value of A (60°) into either Equation 1 or Equation 2 to find B. Using Equation 2:

\[ 60^\circ + B = 90^\circ \]

Subtract 60° from both sides:

\[ B = 90^\circ - 60^\circ \]

\[ B = 30^\circ \]

Step 3: Verify the conditions

We found A = \(60^\circ\) and B = \(30^\circ\). Let's check if they satisfy the given conditions:

• Are A and B positive? Yes, \(60^\circ > 0\) and \(30^\circ > 0\).
• Are A and B acute angles? Yes, \(0^\circ < 60^\circ < 90^\circ\) and \(0^\circ < 30^\circ < 90^\circ\).
• Is \(A \ge B\)? Yes, \(60^\circ \ge 30^\circ\).

All conditions are satisfied by the calculated values of A and B.

Step 4: Compare with options

The calculated measures for A and B are \(60^\circ\) and \(30^\circ\).

Comparing this with the given options:

The measures \(60^\circ\) and \(30^\circ\) match Option 2.

Conclusion

Based on the given trigonometric equations \( \cos (A - B) = \frac{{\sqrt 3 }}{2} \) and \( \cos (A + B) = 0 \), and the conditions that A and B are positive acute angles with \(A \ge B\), the measures of A and B are found to be \(60^\circ\) and \(30^\circ\) respectively.

Revision Table: Key Trigonometric Values

It's helpful to remember common trigonometric values for standard angles when solving such problems.

Additional Information: Trigonometric Identities

While not directly used in this specific solution, identities for \( \cos(A-B) \) and \( \cos(A+B) \) are fundamental in trigonometry.

• The cosine difference identity is \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).
• The cosine sum identity is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).

These identities allow us to express the cosine of a sum or difference in terms of the sines and cosines of the individual angles A and B. In this problem, we used the inverse cosine function (arccos) to find the angles directly from the given values.