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Inverse Trig Functions Principal Solutions

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

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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.

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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

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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: Q3 or Q4? For Arccos & Arcsec: Q2 or Q3? For Arctan & Arccot: Q2 or Q4?

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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 Q4 For Arccos & Arcsec: use Q2 For Arctan & Arccot: use Q4

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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.

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Arcsin/Arccsc l 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 Q3 Q4

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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.

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Arccos/Arcsec l 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/2 π -π/2 Q1Q2Q3 Q4

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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.

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Arctan/Arccot l 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 Q2 Q3 Q4 π/2 π -π/2 -π

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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.

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

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

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

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

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Practice l l l l l l

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