Inverse Trigonometric Functions Notes

Inverse Trigonometric Functions

Evaluating Inverse Trig Functions

  • To find the angle that satisfies a specific trigonometric equation, use the inverse trig functions.

Key Definitions

  • Inverse Sine (Arcsine): If ( ext{sin}( heta) = x ), then ( ext{arcsin}(x) = heta ) is defined for ( -1 \leq x \leq 1 ).

  • Inverse Cosine (Arccosine): If ( ext{cos}( heta) = x ), then ( ext{arccos}(x) = heta ) is defined for ( -1 \leq x \leq 1 ) and ( 0 \leq heta \leq \pi ).

  • Inverse Tangent (Arctangent): If ( ext{tan}( heta) = x ), then ( ext{arctan}(x) = heta ) is defined for all real numbers ( x ).

Evaluating Examples

  1. Evaluate: ( ext{sin}(7/4) )

    • Use the unit circle to find the reference angle in Quadrants I or II where ( y = \frac{3}{2} ).

  2. Evaluate: ( ext{sin}(?) = \frac{\sqrt{3}}{2} )

    • Solution: ( ? = \text{arcsin}(\frac{\sqrt{3}}{2}) ) corresponds to angle ( \frac{\pi}{3} ) in QI.

  3. Angle values for given y-values:

    • For ( y = 0 ): ( ext{arcsin}(0) = 0 )

    • For ( y = \frac{\sqrt{3}}{2} ): ( ext{arcsin}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3} )

    • For ( y = -\frac{\sqrt{3}}{2} ): ( ext{arcsin}(-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3} )

  4. Non-defined Values:

    • ( ext{arcsin}(\frac{3}{2}) ) is undefined as ( x ) must be in the range ( -1 \leq x \leq 1 ).

    • Likewise ( ext{arccos}(-8) ) is undefined.

Graphical Representation

  • Quadrants:

    • In Quadrant I: ( x ) and ( y ) are positive.

    • In Quadrant II: ( x ) is negative, ( y ) is positive.

    • In Quadrant III: both values are negative.

    • In Quadrant IV: ( x ) is positive, ( y ) is negative.

Composition of Trigonometric Functions

  • To evaluate the composition, process the innermost function first, followed by the outer function:

    • Example: ( ext{sin}( ext{arccos}(-\frac{\sqrt{3}}{2})) )

    • Start with ( ext{arccos}(-\frac{\sqrt{3}}{2}) ), then apply ( ext{sin} ).

Additional Evaluations

  • Evaluate Meanings:

    • ( ext{sin}( ext{arcsin}(y)) = y )

    • ( ext{cos}( ext{arccos}(x)) = x )

    • For adding angles with inverse functions, use the reference angle in the appropriate quadrant.

Inverse Function Ranges

  • Range for ( ext{arcsin}(x) ): ( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} )

  • Range for ( ext{arccos}(x) ): ( 0 \leq y \leq \pi )

  • Range for ( ext{arctan}(x) ): ( -\frac{\pi}{2} < y < \frac{\pi}{2} )

Summary

  • Always refer to the unit circle for angles and their corresponding y-values.

  • Remember that inverse functions help find the angle from a given sine, cosine, or tangent value.