Question

If sin x = - 4/5 where pi < x < (3pi)/2 and cos y = 8/17 where 0 < y < n/2 , find any two of the following: either sin(x - y), cos(x - y) , or tan(x - y) .

Answer 1

The answer is

sin(x-y)= 13/85

cos(x-y)= -36/85

What we know.

$\sin(x) = - \frac{4}{5}$

$\cos(y) = \frac{8}{17}$

The intervals mean that all values of x are in the 3rd quadrant. The second one mean that all y values are in the 1st quadrant.

Let find sin( x-y).

$\sin(x - y) = \sin(x) \cos(y) - \cos(x) \sin(y)$

We can find sin of y.

Remeber that all y values are in the 1st quadrant. This means y is positive.

• To find y, we know that cos y=8/17.
• This means that we know the adjacent side/hypotenuse
• Sine is opposite/hypotenuse.
• We can find the opposite side by doing pythagorean theorem.
• ${x}^{2} + {y}^{2} = {r}^{2}$
• ${8}^{2} + {y}^{2} + {17}^{2}$
• $64 + {y}^{2} = 289$
• $y {}^{2} = 225$
• $y = 15$
• So that means sin of y=15/17

We can find cos x by doing the exact opposite.

We know that sin x=-4/5 so we need to find the adjacent/hypotenuse side.

$( { - 4}{} ) {}^{2} + {y}^{2} = 5 {}^{2}$

$16 + {y}^{2} = 25$

${y}^{2} = 9$

$y = 3$

Remember cosine in the 3rd quadrant is negative so

cos x=-3/5.

Plug the values into the earlier formula,

$\sin ( \frac{ - 4}{5} ) \ \cos( \frac{8}{17} ) - \cos( - \frac{3}{5} ) \sin( \frac{15}{17} )$

Multiply the fractions then multiply it

$\frac{ - 32}{85} + \frac{45}{85} = \frac{13}{85}$

For cos(x-y).

Plug in the same ones for earlier,

$\cos(x - y) = \cos(x) \cos(y) + \sin(x) \sin(y)$

$\cos( \frac{ - 3}{5} ) \cos( \frac{8}{17} ) + \sin( - \frac{4}{5} ) \sin( \frac{15}{17} )$

$- \frac{24}{85} - \frac{60}{85} = - \frac { 36}{85}$