Find the exact value of each expression

Question

sin(inverse tan 3)

Answer 1

          /|

         / |

√10  /  |

       /   | 3

      /    |

     /θ__|

        1

 

tanθ = 3 = 3/1

 

The two sides of the triangle are 3 and 1.  The hypotenuse, then, can be found with the Pythagorean Theorem: √(1²+3²) = √10.  

 

So sinθ = 3/√10

 

You may want to rationalize the denominator:

 

sinθ = (3/√10)*(√10/√10) = (3√10)/10

Answer 2

3/√10.

 

How: draw a right triangle in first quadrant with acute angle theta in standard position; this triangle has horizontal leg 1, vertical leg 3, and hypotenuse square root of 10.

 

tan θ = 3/1, i.e arctan 3 = θ.  Now examine said triangle; sin θ = 3/√10.