Math127 Inverse Trig Functions

Inverse Trigonometric Functions

• Inverse notation is used to find the angle corresponding to a given trigonometric value.

• Example:

  • Sine of 30 degrees equals 1/2.

  • Inverse sine asks: what angle has a sine of 1/2? The answer is 30 degrees.

Properties of Inverse Functions

• Inverse functions have restrictions on their domains and ranges:

  • Sine Inverse: Domain is [-1, 1] and the range is [-π/2, π/2]

  • Cosine Inverse: Domain is [-1, 1] and the range is [0, π]

  • Tangent Inverse: Domain is all real numbers and the range is [-π/2, π/2]

Example Values (0 to 90 degrees)

• Key Angles and Ratios:

  • 30 degrees: 1, √3, 2

  • 45 degrees: 1, 1, √2

  • 60 degrees: √3, 1, 2

Example Calculations

• Sine Inverse of √3/2 :

  • Recognize this ratio relates to 60 degrees (√3 on opposite and 2 as hypotenuse).

• Cosine Inverse of √2/2:

  • Equal to 1/√2 after rationalizing; corresponding angle is 45 degrees.

• Tangent Inverse of √3/3

  • Equivalent to tan(30 degrees).

Handling Negative Values

• When dealing with negative values in inverse functions, determine

  • Example: Cosine Inverse of -1/√2 corresponds to 45 degrees, but negative in the second quadrant equals 135 degrees.

Further Examples

• Cosine Inverse of -√3/2:

  • Identifies 30 degrees but in the second quadrant, results in 150 degrees.

Using a Calculator

• Example: Cosine inverse of 0.7 gives a numerical result of approximately 0.795.

• Error Handling: Sine inverse of -1.2 triggers an error because -1.2 exceeds the domain of [-1, 1].

Composite Functions

• Domains in Composition:

  • For functions to produce results, their values must fall within each respective domain.

Example Domains

• Tangent Inverse can accept any real number (e.g., 5.3 gives 5.3 as a result)

• Sine and Cosine Inverse must be within [-1, 1] (e.g., √2 is outside its domain).

Reference Angles

• Understanding Reference Angles:

  • Helpful when working with angles outside the principal range of inverse trig functions.

Example Calculations with Reference Angles

• Sine Inverse of Sine(2π/3): Out of sine's domain, so find reference angle

• Result: Reference angle is π/3, giving a sine of √3/2, leading back to the angle π/3.

Isolating Trig Functions in Equations

• Always strive to isolate the trig function.

Example Breakdown

• Solve:

  1. Divide equation by 2 to isolate cosine inverse.

  2. Multiply to cancel inverse and solve remainder (e.g., x = cosine(π/2) = 0).

Complex Variables

• Combine sine inverses and simplify to find exact values.

• Remember: If the angles make sine or cosine negative, account for quadrants when finalizing answers.