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Inverse Trig Functions

Inverse Trig Functions. Principal Solutions. Principal Solutions. 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.

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Inverse Trig Functions

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  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-1 0.5 = 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. + + Q3 Q4 Q3 Q4 Q2 Q1 π -π/2 3π/2 π/2 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 ?

  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. + + Q4 Q3 Q1 Q2 Q3 Q4 π/2 -π/2 3π/2 π 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?

  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. Q2 Q1 Q3 Q4 -π -π/2 π π/2 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?

  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, principalsolution. Arcsin & Arccsc: -90° to 90°/-π/2toπ/2 Arccos & Arcsec: 0° to 180° /0 to π Arctan & Arccot: -90° to 90°/-π/2toπ/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

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