Question

1. Find sin? if cos?=1/2 and ? terminates in Quadrant IV. 2. Find cos? if sin?=(?2)/2 and ? terminates in Quadrant I. 3. Find tan? if cos?=-1/2 and ? terminates in Quadrant II. 4. Find tan? if sin?=-1 and 0??<2? radians.

Answer 1

1.
`x` in quadrant IV means
`\sin x<0`, so

`\cos^2x+\sin^2x=1\implies\sin x=-√(1-\cos^2x)=-\frac{\sqrt3}2`

2.
`x` in quadrant I means
`\cos x>0`. Then

`\cos x=√(1-\sin^2x)=\frac{\sqrt2}2`

3.
`x` in quadrant II means
`\sin x>0`. Then

`\tan x=(\sin x)/(\cos x)=(√(1-\cos^2x))/(-\frac12)=\frac{\frac{\sqrt3}2}{-\frac12}=-\sqrt3`

4. If
`\sin x=-1`, then
`\cos x=0`, so
`\tan x` is undefined.