Inverse Trig Functions

Principal Solutions

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.)

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 radiansSin-1 0.5 = 30° or π/6 radians

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?

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

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.

Arcsin/Arccsc

Choose adjacent quadrants with positive & negative y-values :

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 ?

+ +

π/2 π 3π/2-π/2

Q1 Q2 Q3Q3 Q4Q4

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.

Arccos/Arcsec

Choose adjacent quadrants with positive & negative y-values :

Which quadrant of angles is adjacent to Q1, but with negative y-values? What range of solutions is valid?

+ +

π/2 3π/2π-π/2

Q1 Q2 Q3Q3 Q4Q4

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.

Arctan/Arccot

Choose adjacent quadrants with positive & negative y-values :

Which quadrant of angles is adjacent to Q1, over a continuous section, but with negative y-values? What range of solutions is valid?

Q1 Q2Q3 Q4

π/2 π-π/2-π

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.

Practice

Arcsin (-0.5) Arctan 0 Arccos (-1)

Summary - Part 1

Find the one, principal solution.

Arcsin & Arccsc: -90° to 90° / -π/2 to π/2

Arccos & Arcsec: 0° to 180° / 0 to πArctan & Arccot: -90° to 90° / -π/2 to

π/2

Compound Expressions #1

Evaluate:(Start inside the parentheses.)

sin Arc tan 3 Arc cos 3

2

sin Arc tan 3 Arc cos 3

2

sin 60 150 sin(210 )

12

Compound Expressions #2

Evaluate. arcsin sin76

arcsin 1

2

76

2k; 11

6 2k

NOTE: We cannot forget to include all relevant solutions and all of their co-terminal angles.

Practice

tan Arc tan( 1)

Arc cos tan54

arcsin cos Arc sin 3

2