Question

If cos Θ = -4/7 , what are the values of sin Θ and tan Θ?

Answer 1

Sin squared plus cosin squared = 1

Answer 2

\(\bf cos(\theta)=-\cfrac{4}{7}\to \cfrac{-4}{7}\to \cfrac{adjacent}{hypotenuse}\to \cfrac{a=-4}{c=7}
\\ \quad \\
c^2=a^2+{\color{blue}{ b}}^2\implies \pm\sqrt{c^2-a^2}={\color{blue}{ b}}
\\ \quad \\
recall\implies sin(\theta)=\cfrac{{\color{blue}{ b}}}{c}\qquad tan(\theta)=\cfrac{{\color{blue}{ b}}}{a}\)

Answer 3

But how do you know if sin is positive or negative?

Answer 4

\(\Large \color{red}\sin \color{red}\theta = \sqrt{1 -cos^2\theta} =\sqrt{1-(\frac{-4}{7})^2} =\frac{\sqrt{33}}{7}\)
\(\Large \color{red}\tan\color{red} \theta=\huge \frac{sin\theta}{cos\theta} =\frac{\frac{\sqrt{33}}{7}}{\frac{-4}{7}} = \frac{-\sqrt{33}}{4}\)