Question

How do you find trigonometric ratios of 30, 45, and 60 degrees?

Answer 1

The trigonometric ratios for `30^o`, `45^o`, and `60^o` are based on some standard triangles. sin, cos, and tan (and their reciprocals) are the ratios of the sides of these triangles.

Explanation:

Both `30^o` and `60^o` are based on an equilateral triangle with sides of length 2 and with one of the angles bisected.

The `45^o` angle is based on an isosceles triangle with the equal sides having a length of 1.

For all triangles the Pythagorean Theorem is used to compute the "missing" side length.

If remember that
`color(white)("XXXX")`sin `= "opposite"/"hypotenuse"`

`color(white)("XXXX")`cos `= "adjacent"/"hypotenuse"`

`color(white)("XXXX")`tan `= "opposite"/"adjacent"`

and their reciprocals.

Then, for example:
`color(white)("XXXX")` `sin(60^o) = sqrt(3)/2`

`color(white)("XXXX")` `sin(30^o) = 1/2`

`color(white)("XXXX")` `sin(45^o) = 1/sqrt(2)`

`color(white)("XXXX")` `cos(60^o) = 1/2`

etc.

Answer 2

The other answer is fine. I'll just point out that once the student has internalized these Two Tired Triangles of Trig (30/60/90 and 45/45/90) they're done. If you review the trig questions here you'll find the vast majority use just these triangles.