Inverse Trig Functions Principal Solutions

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°.

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.

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-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 : + 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 ?

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 : + 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?

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 : 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?

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 Arcsec 2 Arccot √3 Arccos (-1) Arccsc (-1)

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

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

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

Practice