Question

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?

Answer 1

`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.