Question

Find sin(x – y), given that sin x = ${\displaystyle \frac{8}{17}}$ with 0 < x < ${\displaystyle \frac{\pi }{2}}$, and cos y = ${\displaystyle -\frac{24}{25}}$, x < y < ${\displaystyle \frac{3\pi }{2}}$

Answer 1

sin x = ${\displaystyle \frac{8}{17}}$, 0 < x < ${\displaystyle \frac{\pi }{2}}$

⇒ Where x is in I quadrant

∴ sin x, cos x are +ve

From ΔABX,

AX = ${\displaystyle \sqrt{{17}^2-8^2}}$

= ${\displaystyle \sqrt{(17+8)(17-8)}}$

= ${\displaystyle \sqrt{\left(25\right)\left(9\right)}}$

= 5 × 3

= 15

∴ sin x = ${\displaystyle \frac{8}{17}}$ and cos x = ${\displaystyle \frac{15}{17}}$

cos y = ${\displaystyle -\frac{24}{25},\pi <y<\frac{3\pi }{2}}$

⇒ Where y is in III quadrant

So, sin y and cos y are –ve

From ΔALY,

AL = ${\displaystyle \sqrt{{25}^2-{24}^2}}$

= ${\displaystyle \sqrt{49}}$

= 7

∴ cos y = ${\displaystyle -\frac{24}{25}}$ and sin y = ${\displaystyle -\frac{7}{25}}$

sin(x – y) = sin x cos y – cos x sin y

= ${\displaystyle \left(\frac{8}{17}\right)(-\frac{24}{25})-\left(\frac{15}{17}\right)(-\frac{7}{25})}$

= ${\displaystyle -\frac{192}{425}+\frac{105}{425}}$

= ${\displaystyle -\frac{87}{425}}$