Given: sin A=2/3, A<90° and cosB=(√5)/6, B>90°, determine without solving for A or B, the value of sin(B−A).

To find the value of \(\sin(B - A)\) without solving for \(A\) or \(B\), we can use the trigonometric identity for the sine of a difference:

\[
\sin(B - A) = \sin B \cos A - \cos B \sin A
\]

Given:
- \(\sin A = \frac{2}{3}\), with \(A < 90^\circ\)
- \(\cos B = \frac{\sqrt{5}}{6}\), with \(B > 90^\circ\)

First, we determine \(\cos A\) using the Pythagorean identity:

\[
\cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}
\]
\[
\cos A = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
\]

Since \(A < 90^\circ\), \(\cos A\) is positive, so \(\cos A = \frac{\sqrt{5}}{3}\).

Next, we determine \(\sin B\) using the Pythagorean identity:

\[
\sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{\sqrt{5}}{6}\right)^2 = 1 - \frac{5}{36} = \frac{31}{36}
\]
\[
\sin B = \sqrt{\frac{31}{36}} = \frac{\sqrt{31}}{6}
\]

Since \(B > 90^\circ\), \(\sin B\) is positive and \(\cos B\) is negative. Thus, \(\sin B = \frac{\sqrt{31}}{6}\) and \(\cos B = -\frac{\sqrt{5}}{6}\).

Now, substitute the values into the identity:

\[
\sin(B - A) = \sin B \cos A - \cos B \sin A
\]
\[
\sin(B - A) = \left(\frac{\sqrt{31}}{6}\right)\left(\frac{\sqrt{5}}{3}\right) - \left(-\frac{\sqrt{5}}{6}\right)\left(\frac{2}{3}\right)
\]

Simplify each term:

\[
\sin(B - A) = \frac{\sqrt{155}}{18} + \frac{2\sqrt{5}}{18}
\]

Combine the terms:

\[
\sin(B - A) = \frac{\sqrt{155} + 2\sqrt{5}}{18}
\]

This is the value of \(\sin(B - A)\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you derive the Pythagorean identity used for \(\cos A\) and \(\sin B\)?
2. What would change in the calculation if \(A\) or \(B\) were in different quadrants?
3. Can you find \(\cos(B - A)\) using a similar identity?
4. How would the result change if \(B < 90^\circ\) and \(A > 90^\circ\)?
5. Can you use the same method to find \(\tan(B - A)\)?

**Tip:** Remember that the sign of trigonometric functions changes based on the quadrant of the angle. Always consider the angle's range when calculating values.

Learn how to calculate \(\sin(B - A)\) using given trigonometric values without solving for \(A\) or \(B\). Explore the application of trigonometric identities and Pythagorean identity in solving advanced trigonometry problems.

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