Question

If tan Ф = 3/4 , calculate 3 sinФ +2cos Ф

Answer 1

Recall that

`\sin^2\phi+\cos^2\phi=1\implies\tan^2\phi+1=\sec^2\phi`

Then with
`\tan\phi=\frac34`, we have

`\sec^2\phi=\frac9{16}+1=(25)/(16)\implies\sec\phi=\pm\frac54\implies\cos\phi=\pm\frac45`

Since

`\tan\phi=(\sin\phi)/(\cos\phi)=(\sin\phi)/(\pm\frac45)=\frac34`

it follows that

`\sin\phi=\pm\frac35`

So,

`3\sin\phi+2\cos\phi=\pm\frac95\pm\frac85=\pm\frac{17}5`