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

2 Answers
Jun 25, 2015

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.
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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.

Jun 4, 2018

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.