With reference to the right-angled triangle shown, what is the value of $\sin(A)\cos(B) + \cos(A)\sin(B)$?
To solve the given problem, we need to evaluate the expression: \(\sin(A)\cos(B) + \cos(A)\sin(B)\).
This expression resembles the formula for the sine of the sum of two angles. According to trigonometric identities:
\(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\)
Thus, the given expression is simply \(\sin(A + B)\).
In a right-angled triangle, the sum of angles \(A\) and \(B\) equals 90 degrees since the third angle is 90°.
\(A + B = 90^\circ\)
Therefore, \(\sin(A + B) = \sin(90^\circ)\).
We know from trigonometric values that:
\(\sin(90^\circ) = 1\)
Thus, the value of \(\sin(A)\cos(B) + \cos(A)\sin(B)\) is 1.
Therefore, the correct answer is 1.
Important Questions from Trigonometry (Notes)
- If $\tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} = ?$
- $\tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} = ?$
- For which values of $A$ and $B$ is $\sin A = \cot B$?
- Which of the following best approximates $\sin(0.5^\circ)$?
- Which one of the following graphs represents $f(x) = \sin x \cos x$?
