Working with Trig Inverses
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center


When working with trigonometric inverse problems, remember the following information:

  Inverse Notation: 

arcsin(x) = sin-1(x)
arccos(x) = cos-1(x)
arctan(x) = tan-1(x)
arcsin(x) is read "the angle whose sine is x".

  Reference Triangles:

 

Examples:

You are looking for "the angle whose sine is" and the angle is in the interval  .

and we know that ,  so or 45º.


Sine is positive in QI.

 
 

You are looking for the sine of "an angle whose cosine is" 1/2 in the interval .

and we know that , so .  Now, finish the question by finding the sine of this result:
                    


Cosine is positive in QI.



 

You are looking for "the angle whose tangent is" and the angle is in the interval  .

and we know that a reference angle of 30 degrees in quadrant IV will set up our situation.  We need to remember that
.



Tangent is negative in QIV.


 


 

You are looking for "the angle whose tangent is" and the angle is in the interval  .
creates the triangle shown at the right.  The Pythagorean Theorem was used to find the length of the hypotenuse.



Tangent is positive in QI.


 

You are looking for "the angle whose sin is" the same as the in the interval  .

You need to find an angle whose in . 

Sine is positive in QI.


 

Use the inverse function of sine, cosine, or tangent to solve this problem:
Given:   
and ,  find the exact value of .

To solve this problem we need to know the "angle whose" cotangent is
-8/15, which would imply the use of the inverse of cotangent.  But, our question specifies the use of the inverse of sine, cosine or tangent only.  Let's rewrite the first given condition to be:
and use  
Our problem is to now find



Since this question asks for "exact value", the calculator cannot be used to give a decimal approximation.

Answer:    


 

Tough one!!!!

You are looking for secant of "the angle whose sin is" 2x-1 in the interval  .
From labeling the triangle and using the Pythagorean Theorem, we can read the secant of the angle.

 

Use the TI-83+/84+
graphing calculator
to investigate trig
inverses.
Click here.