Question

How do you sketch a right triangle corresponding to `sectheta=2` and find the third side, then find the other five trigonometric functions?

Answer 1

See below:

Explanation:

With a triangle with `sectheta=2`, let's first remember that:

`sectheta="hypotenuse"/"adjacent"=2/1`

We could run through the pythagorean theorem for the third side, but we can also remember that these two measurements are part of a 30-60-90 triangle, and so the third side is `sqrt3`. To prove it, let's go ahead and do Pythagorean Theorem:

`a^2+b^2=c^2`

`1^2+(sqrt3)^2=2^2`

`1+3=4`

This gives us the 6 trig ratios:

`sintheta=sqrt3/2`

`costheta=1/2`

`tantheta=(sqrt3/2)/(1/2)=(sqrt3/2)(2/1)=sqrt3`

`csctheta=2/sqrt3=(2sqrt3)/3`

`sectheta=2/1=2`

`cottheta=1/sqrt3=sqrt3/3`

The angle of the triangle we're looking at is `60=pi/3`, with the opposite being the middle length of `sqrt3`, the adjacent length of 1, and hypotenuse of 2.