# How do you find the five remaining trigonometric function satisfying sintheta=3/8, costheta<0?

Jun 4, 2017

cosΘ=(-√55)/8
tanΘ=(-3√55)/55
cscΘ=8/3
secΘ=(-8√55)/55
cotΘ=(-√55)/3

#### Explanation:

Since $\sin$ is positive and $\cos$ is less than 0, it means that this triangle is in quadrant 2.

Use Pythagorean's theorem (${a}^{2} + {b}^{2} = {c}^{2}$) and plug in your given values: $a = 3$ and $c = 8$ and solve.

${3}^{2} + {b}^{2} = {8}^{2}$$9 + {b}^{2} = 64$${b}^{2} = 55$b=-√55 (it is negative because it is in quadrant 2 and the $\cos$ is the x value of the triangle)

Now that you have all side lengths, simply plug them in to the remaining trigonometric functions:

cosΘ=(-√55)/8
tanΘ=3/(-√55)=(-3√55)/55
cscΘ=8/3
secΘ=8/(-√55)=(-8√55)/55
cotΘ=(-√55)/3