How do you find the values of all six trigonometric functions of a right triangle ABC where C is the right angle, given a=1, b=3?

1 Answer
May 7, 2018

#sin=(3sqrt10)/10#
#csc=sqrt10/3#
#cos=sqrt10/10#
#sec=sqrt10/1#
#tan=3#
#cot=1/3#
Note: This is based on my interpretation of the diagram. Without more information it is impossible to know which side, #a# or #b#, is adjacent or opposite.

Explanation:

First, finish solving the reference triangle, using the Pythagorean Theorem:
#a^2+b^2=c^2# Input knowns:
#(1)^2+(3)^2=c^2# Simplify:
#1+9=c^2# Isolate #c#:
#c=sqrt10# Since #c# is the hypotenuse of a reference triangle it must be positive.

Using the definitions of trig functions (assuming #a# is "adjacent" and #b# is "opposite"):
#sin=o/h=b/c=3/sqrt10=(3sqrt10)/10#
#csc=h/o=c/b=sqrt10/3#
#cos=a/h=a/c=1/sqrt10=sqrt10/10#
#sec=h/a=c/a=sqrt10/1#
#tan=o/a=b/a=3/1=3#
#cot=a/o=a/b=1/3#
Note: This is based on my interpretation of the diagram. Without more information it is impossible to know which side, #a# or #b#, is adjacent or opposite.