1. Inverse Trig Functions
Principal Solutions

2. Principal Solutions
In the last section we saw that an INVERSE TRIG function has infinite solutions:
arctan 1 = 45° + 180k
But there is only one PRINCIPAL SOLUTION, 45°.

3. Principal Solutions
Each inverse trig function has one set of Principal Solutions. (If you use a calculator to evaluate an inverse trig function you will get the principal solution.)
We give the Principal Solution when the inverse trig function is capitalized, Arcsin or Sin-1.

4. But Which Solution?
If you are evaluating the inverse trig function of a positive number, it probably won’t surprise you that the principal solution is the Quadrant I angle:
Arctan 1 = 45° or π/4 radians
Sin = 30° or π/6 radians

5. Negative Numbers?
But if you are evaluating the inverse trig function of a negative number, you must decide which quadrant to use.
For Arcsin & Arccsc: Q3 or Q4?
For Arccos & Arcsec: Q2 or Q3?
For Arctan & Arccot: Q2 or Q4?

6. The Right Choice
There is a clear set of rules regarding which quadrants we choose for principal inverse trig solutions:
For Arcsin & Arccsc: use Q4
For Arccos & Arcsec: use Q2
For Arctan & Arccot: use Q4

7. But WHY?
The choice of quadrants for principal solutions was not made without reason. The choice was made based on the graph of the trig function. The next 3 slides show the justification for each choice.

8. Arcsin/Arccsc
Choose adjacent quadrants with positive & negative y-values : +
π/2 π
3π/2
-π/2 Q1 Q2 Q3 Q4
Q3 and 4 are not adjacent to Q1, unless we look to the left of the y-axis. Which angles in Q4 are adjacent to Q1 ?

9. Arcsin/Arccsc
Principal Solutions to Arcsin must be between -90° and 90° or - π/2 and π/2 radians, that includes Quadrant IV angles if the number is negative and Quadrant I angles if the number is positive.

10. Arccos/Arcsec
Choose adjacent quadrants with positive & negative y-values : +
π/2
3π/2 π
-π/2 Q1 Q2 Q3 Q4
Which quadrant of angles is adjacent to Q1, but with negative y-values? What range of solutions is valid?

11. Arccos/Arcsec
Principal Solutions to Arccos must be between 0° and 180° or 0 and π radians, that includes Quadrant II angles if the number is negative and Quadrant I angles if the number is positive.

12. Arctan/Arccot
Choose adjacent quadrants with positive & negative y-values : Q1 Q2 Q3 Q4
π/2 π
-π/2 -π
Which quadrant of angles is adjacent to Q1, over a continuous section, but with negative y-values? What range of solutions is valid?

13. Arctan/Arccot
Principal Solutions to Arctan must be between -90° and 90° or -π/2 and π/2 radians, that includes Quadrant IV angles if the number is negative and Quadrant I angles if the number is positive.

14. Practice Arcsin (-0.5) Arctan 0 Arcsec 2 Arccot √3 Arccos (-1)
Arccsc (-1)

15. Summary - Part 1
If the inverse trig function begins with a CAPITAL letter, find the one, principal solution.
Arcsin & Arccsc: -90° to 90° / -π/2 to π/2
Arccos & Arcsec: 0° to 180° / 0 to π
Arctan & Arccot: -90° to 90° / -π/2 to π/2

16. Compound Expressions #1
Evaluate:
(Start inside the parentheses.)

17. Compound Expressions #2
Evaluate.
NOTE: We cannot forget to include all relevant solutions and all of their co-terminal angles.

18. Practice