Question

Without using a calculator determine the exact value of sin(x - y) given that sin x = -1/3) , tan y = (1/3) , x is a quadrant II angle and y is a quadrant III angle.

Answer 1

sin(x) shouldn't be negative if x is in the second quadrant. Is there a typo?

Answer 2

Im sorry the problem is

Without using a calculator determine the exact value of cos(x - y) given that sin x = (1/3) , tan y = (1/3) , x is a quadrant II angle and y is a quadrant III angle.

Answer 3

its easy

Answer 4

use sum/diff of angle expansion

Answer 5

= cos(x)cos(y)+sin(x)sin(y)

Answer 6

if sin(x) = (1/3) , draw up a general right angle triangle , mark an angle x ( obviously not the right angle ) , and label the opposite 1 and hypotenuse 3

Answer 7

then you can find the adjacent side by pythagorus , which works out as sqrt(8)

therefore cos(x) = - sqrt(8) / 3 ( because x is negative in the 2nd quadrant

Answer 8

now you just need to find siny and cosy

Answer 9

Possible answers are
\[A. 0 B. (12\sqrt{5} - \sqrt{10}) /30 C. (-12\sqrt{5} - \sqrt{10}) /30 D. (-12\sqrt{5} + \sqrt{10}) /30\] \[E. (12\sqrt{5} + \sqrt{10}) /30\]

Answer 10

with tan(y)=(1/3) , draw up a general right angle triangle , mark opposite as 1 and adjacent as 3

Answer 11

therefore the hypotenuse is sqrt(10)

so sin(y) = - 1 / sqrt(10) and cosy =- 3/sqrt(10)

Answer 12

then sub in

Answer 13

So my equation should look like

(-sqrt(8)/3 * -3/sqrt(10) ) + (1/3 * -1/sqrt(10))

???