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

2 Answers
Alan P.
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.

Dean R.
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.

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