Inverse Trig Functions Principal Solutions Principal Solutions l

Inverse Trig Functions Principal Solutions

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

Principal Solutions l 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. ) l We give the Principal Solution when the inverse trig function is capitalized, Arcsin or Sin-1.

But Which Solution? l 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? l But if you are evaluating the inverse trig function of a negative number, you must decide which quadrant to use. • For Arcsin & Arccsc: Q 3 or Q 4? • For Arccos & Arcsec: Q 2 or Q 3? • For Arctan & Arccot: Q 2 or Q 4?

The Right Choice l There is a clear set of rules regarding which quadrants we choose for principal inverse trig solutions: • For Arcsin & Arccsc: use Q 4 • For Arccos & Arcsec: use Q 2 • For Arctan & Arccot: use Q 4

But WHY? l 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 l Choose adjacent quadrants with positive & negative y-values : Q 3 Q 4 -π/2 Q 1 + + π/2 Q 3 Q 2 π Q 4 3π/2 Q 3 and 4 are not adjacent to Q 1, unless we look to the left of the y-axis. Which angles in Q 4 are adjacent to Q 1 ?

Arcsin/Arccsc l 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 l Choose adjacent quadrants with positive & negative y-values : + Q 4 Q 3 -π/2 + Q 1 Q 2 π/2 Q 3 π Q 4 3π/2 Which quadrant of angles is adjacent to Q 1, but with negative y-values? What range of solutions is valid?

Arccos/Arcsec l 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 l Choose adjacent quadrants with positive & negative y-values : Q 3 -π Q 4 -π/2 Q 1 π/2 π Which quadrant of angles is adjacent to Q 1, over a continuous section, but with negative yvalues? What range of solutions is valid?

Arctan/Arccot l 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 l l l Arcsin (-0. 5) Arctan 0 Arcsec 2 Arccot √ 3 Arccos (-1) Arccsc (-1)

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

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

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

Practice l l l