Answer:
[tex]\mathsf {sin (A - B) =\frac{4}{5} }[/tex]
Step-by-step explanation:
[tex]\textsf {Trigonometric Identities to keep in mind :}[/tex]
- [tex]\mathsf {sin (A - B) = sinAcosB - cosAsinB}[/tex]
- [tex]\mathsf {sin \theta = opposite/hyporenuse}[/tex]
- [tex]\mathsf {cos \theta = adjacent/hypotenuse}[/tex]
[tex]\textsf {To find cos A and cos B, we can use the Pythagorean Theorem} \\\textsf {to find the missing sides so we can take the ratio.}[/tex]
[tex]\textsf {Finding the adjacent side of angle A :}[/tex]
[tex]\implies \mathsf {24^{2} + x^{2} = 25^{2} }[/tex]
[tex]\implies \mathsf {x^{2} = 625 - 576 }[/tex]
[tex]\implies \mathsf {\sqrt{x^{2} } = \sqrt{49} }[/tex]
[tex]\implies \mathsf {x = 7}[/tex]
[tex]\textsf {Hence, cos A will be :}[/tex]
[tex]\implies \textsf {cosA = 7/25 (as cos is positive in the 1st Quadrant)}[/tex]
[tex]\textsf {Finding the adjacent side of angle B :}[/tex]
[tex]\implies \mathsf {4^{2} + x^{2} = 5^{2} }[/tex]
[tex]\implies \mathsf {x^{2} = 25 - 16 }[/tex]
[tex]\implies \mathsf {\sqrt{x^{2} } = \sqrt{9} }[/tex]
[tex]\implies \mathsf {x = 3}[/tex]
[tex]\textsf {Hence, cos B will be :}[/tex]
[tex]\implies \textsf {cos B = 3/5 (as cos is positive in the 4th quadrant)}[/tex]
[tex]\textsf {Finding sin (A - B) :}[/tex]
[tex]\implies \mathsf {sin (A - B) = (\frac{24}{25} \times \frac{3}{5}) - (\frac{7}{25} \times -\frac{4}{5}) }[/tex]
[tex]\implies \mathsf {sin (A - B) = \frac{72}{125} + \frac{28}{125}}[/tex]
[tex]\implies \mathsf {sin (A - B) = \frac{100}{125} = \frac{4}{5} }[/tex]
