1. Inverse Trig = Solve for the Angle
Inverse Trig Functions
Inverse Trig = Solve for the Angle

2. INVERSE vs. RECIPROCAL
Its important not to confuse an INVERSE trig function with a RECIPROCAL trig function.
The reciprocal of SINE is COSECANT: (sin x ) -1 = csc x
The inverse of SINE is ARCSIN: sin -1 x = arcsin x
The notation is very important - be careful !!

3. Evaluating INVERSE Trigs
Evaluating an inverse trig means finding the ANGLE that gives us the stated ratio: For example, to evaluate arcsin 0.5, (read “the arcsine of 0.5”) we must ask ourselves “for what angle does sin = 0.5 ? “
Answer: 30° or π/6 radians

4. How Many Answers are Right?
There are other angles for which the sine of that angle = For example
sin 30° = 0.5
sin 150° = 0.5
sin 390° = 0.5
The way we handle this is to give a general solution, that is written in a way such that all possible solutions are included in it.

5. General Solutions
Your solution can include all angles that are co-terminal to the solution angle by adding 360k or 2πk to the angle, where k is any integer (positive or negative). In our example, the first obvious angle was 30° or π/6 radians: arcsin 0.5 = 30° + 360k

6. General Solutions
To be complete, the solution must include angles from other quadrants that produce the same sign and value. Think of this as the reverse of the process we used when we found reference angles --- this time we know the reference angle is 30° and the sign is positive, so we have to find the angle in quadrant II (the other quadrant where sine is positive) that shares the same reference angle (30°). The answer ?
150°

7. General Solutions Our complete, general solution is: Solve these:
arcsin 0.5 = 30° + 360k ; 150° + 360k OR, in radians
arcsin 0.5 = π/6 + 2πk ; 5π/ k
Solve these:
arccos (-0.5)
arcsec 1

8. General Solutions Solutions:
arccos (-0.5) = 120° + 360k; 240° + 360k
OR arccos (-0.5) = 2π/3 + 2πk; 4π/3 + 2πk
arcsec 1 = 0° + 360k OR arcsec 1 = 0 + 2πk
Notice that in our second example there is only one angle between 0° and 360° where secant = 1. The secant of 180° = -1.

9. The Briefest Answer Evaluate arctan (-1 )
First find the Quad I angle where tan = 1: 45° or π/4 radians
Now, in which quadrants is tan negative ? Quadrants II and IV
Angles: 135° + 360k ; 315° + 360k But this answer can be simplified to 135° + 180k Why does this satisfy ALL the angles ?

10. Practice Problems arcsin (-1 ) arccos (√2/2 ) arctan (-√3 ) arccsc √2
arccot 0
arcsec (-2 )

11. Summary
When evaluating a trig function for a given angle, there is ONE ANSWER. Remember all the rules for evaluating trig functions: reference & coterminal angles, positive & negative quadrants, etc.
When evaluating an inverse trig function, there can be INFINITE ANSWERS.

12. Summary
First find the angle in quadrant I that yields the desired ratio (the number given in the inverse trig function)
Then determine which quadrant(s) the solution is in to achieve the given sign - sometimes there is only 1 angle, that is OK
State the angle answer(s) for those quadrants

13. Summary
Add a multiple of k to include all co-terminal solutions (i.e k or + 2πk ) for each of your angle solutions
Finally, it may be possible to make your answer more brief, like we did with arctan, this is usually the case when the solutions differ by exactly 180° or π radians