Inverse Trigonometric functions

Inverse Functions of Trigonometric Ratios

Introduction

  • Inverse trigonometric functions are used to find interior angles of a right triangle when two sides are known.

  • If one angle (e.g., 50 degrees) in a right triangle is known, the third angle can be easily found since the sum of angles in a right triangle is 180 degrees, and one angle is already 90 degrees. So, the other angle is 90 - known angle.

  • Trigonometry is primarily used when only the sides of a right triangle are known, and an angle needs to be determined.

  • If all sides are unknown except for angle, Pythagorean theorem (a^2 + b^2 = c^2) can be used.

Determining Angles Using Inverse Trig Functions

  • The goal is to find an unknown angle by relating it to known sides of a right triangle.

  • Consider a right triangle with angle X, opposite side of 200 feet, and adjacent side of 50 feet.

  • Using SOH CAH TOA, identify the trigonometric function that relates the opposite and adjacent sides which is tangent.

    tan(X) = \frac{opposite}{adjacent} = \frac{200}{50}

Applying Inverse Tangent

  • To isolate angle X, apply the inverse tangent function to both sides of the equation.

    tan^{-1}(tan(X)) = tan^{-1}(\frac{200}{50})

  • The inverse tangent cancels out the tangent on the left side, leaving X.

    X = tan^{-1}(\frac{200}{50})

  • Simplifying the fraction:

    X = tan^{-1}(4)

Calculator Usage

  • Calculators have inverse trigonometric functions, often denoted as tan^{-1} or arctan.

  • The method to access these functions depends on the calculator model.

    • For some calculators (e.g., TI-34), it may be accessed through the "second" button followed by a trig function option.

    • For TI-84, press "second" then the tangent button.

  • Make sure the calculator is in degree mode to obtain the angle in degrees and not radians.

  • Calculating the example:

    X = tan^{-1}(4) ≈ 75.96 degrees

Additional Example

  • Consider three buildings with a right triangle formed between their centers.

  • The distance between two buildings (hypotenuse) is 300 feet, and the vertical distance (opposite) is 120 feet.

  • To find the angle X, use the sine function:

    sin(X) = \frac{opposite}{hypotenuse} = \frac{120}{300}

  • Apply the inverse sine to both sides:

    X = sin^{-1}(\frac{120}{300})

  • Using a calculator:

    X ≈ 23.58 degrees

So the angle is approximately 23.58 degrees.

Summary

  • Identify the appropriate trigonometric ratio (sine, cosine, tangent) based on the known sides.

  • Set up the equation relating the angle to the sides.

  • Apply the inverse trigonometric function to both sides to solve for the angle.

  • Use a calculator in degree mode to find the angle value.

  • Practice these problems to master them.