Find the exact values of the six trigonometric functions of 𝜃.

Question

Find the exact values of the six trigonometric functions of 𝜃.

sin 𝜃	=			csc 𝜃	=
cos 𝜃	=			sec 𝜃	=
tan 𝜃	=			cot 𝜃	=

Answer 1

The point on the circle is (35/37, 12/37) so we can draw the triangle like this:

To determine the value of "h", use the Pythagorean Theorem: a² + b² = c² so:

h = √(a² + b²)

h = √[(35/37)² + (12/37)²]

h = √[(1225/1369) + (144/1369)]

h = √[(1225 + 144)/1369]

h = √[1369/1369] = 1

(Note that we really didn't need to do this in this case because the drawing shows the circle goes through the point (1, 0) so obviously the radius is 1. I did this, though, so that you can see how to do it if the radius is not shown.)

sin(θ) = opp/hyp = (12/37)/1 = 12/37

cos(θ) = adj/hyp = (35/37)/1 = 35/37

tan(θ) = opp/adj = (12/37)/(35/37) = (12/37)•(37/35) = 12/35

csc(θ) = hyp/opp = 1/(12/37) = 1•(37/12) = 37/12

sec(θ) = hyp/adj = 1/(35/37) = 1•(37/35) = 37/35

cot(θ) = adj/opp = (35/37)/(12/37) = (35/37)•(37/12) = 35/12