Solving Trigonometry Angles: Finding A + B

This problem asks us to find the sum of two angles, A and B, given the sine of angle A and the cosine of angle B. We will use our knowledge of standard trigonometric values to find the individual angles A and B, and then sum them up.

Step-by-Step Solution

We are given the following information:

• \( \sin A = \frac{1}{{\sqrt 2 }} \)
• \( {\mathop{\rm Cos}\nolimits} B = \frac{{\sqrt 3 }}{2} \)

Let's find the value of angle A first.

We know the standard trigonometric values. The angle whose sine is \( \frac{1}{{\sqrt 2 }} \) is \( 45^\circ \).

Therefore, \( A = 45^\circ \).

Now, let's find the value of angle B.

We look for the angle whose cosine is \( \frac{{\sqrt 3 }}{2} \). From standard trigonometric values, we know that this angle is \( 30^\circ \).

Therefore, \( B = 30^\circ \).

Finally, we need to find the value of \( (A + B)^\circ \).

Substitute the values of A and B we found:

\( A + B = 45^\circ + 30^\circ \)

\( A + B = 75^\circ \)

So, the value of \( (A + B)^\circ \) is \( 75^\circ \).

Summary of Trigonometric Values

It's helpful to remember common trigonometric values for angles like \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \). Here is a quick reference:

Using this table confirms that \( \sin 45^\circ = \frac{1}{{\sqrt 2 }} \) and \( \cos 30^\circ = \frac{{\sqrt 3 }}{2} \).

Conclusion

By determining angles A and B from the given sine and cosine values using standard trigonometric ratios, we calculated their sum. Angle A is \( 45^\circ \) and angle B is \( 30^\circ \), leading to a sum of \( 75^\circ \).

Revision Table: Trigonometry Basics

Additional Information: Inverse Trigonometric Functions

The process of finding an angle from its sine or cosine value is related to inverse trigonometric functions.

• If \( \sin A = x \), then \( A = \sin^{-1}(x) \) or \( A = \arcsin(x) \). This means A is the angle whose sine is x.
• If \( \cos B = y \), then \( B = \cos^{-1}(y) \) or \( B = \arccos(y) \). This means B is the angle whose cosine is y.

In this problem:

• \( A = \sin^{-1}\left(\frac{1}{{\sqrt 2 }}\right) \). For angles between \( 0^\circ \) and \( 90^\circ \), this gives \( A = 45^\circ \).
• \( B = \cos^{-1}\left(\frac{{\sqrt 3 }}{2}\right) \). For angles between \( 0^\circ \) and \( 90^\circ \), this gives \( B = 30^\circ \).

Inverse trigonometric functions are crucial when the angle is not one of the standard values or when dealing with trigonometric equations.