Question

Angles θ and φ are angles in standard position such that:

sinθ = -5/13 and θ terminates in Quadrant III

tanφ = -8/15 and φ terminates in Quadrant II


Find sin(θ + φ).

Answer 1

When
`\theta` terminates in quadrant III, both
`\cos\theta` and
`\sin\theta` are negative, and

`\sin^2\theta+\cos^2\theta=1\implies\cos\theta=-√(1-\sin^2\theta)=-(12)/(13)`

When
`\varphi` terminates in quadrant II,
`\cos\varphi` is negative and
`\sin\varphi` is positive, so

`1+\tan^2\varphi=\sec^2\varphi\implies\sec\varphi=-(17)/(15)`

which gives

`\cos\varphi=\frac1{-(17)/(15)}=-(15)/(17)`

`\tan\varphi=(\sin\varphi)/(\cos\varphi)=-\frac8{15}\implies\sin\varphi=\frac8{17}`

Now,

`\sin(\theta+\varphi)=\sin\theta\cos\varphi+\cos\theta\sin\varphi=-(21)/(221)`