Evaluating inverse trigonometric functions

Evaluating inverse trigonometric functions

Lessons

In this lesson, we will learn:

  • Application of the Cancellation Laws
  • Solving Expressions With One Inverse Trigonometry
  • Evaluating Expressions With a Combination of Inverse and Non-Inverse Trigonometry
  • Special Cases: Evaluating Functions With Numbers Outside of the Restrictions

Cancellation Laws:

sin1(sinx)=x\sin^{-1} (\sin x) = x\;, π2xπ2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}

sin(sin1x)=x\sin (\sin^{-1} x) = x\;, 1x1-1 \leq x \leq 1

cos1(cosx)=x\cos^{-1} (\cos x) = x\;, 0xπ0 \leq x \leq \pi

cos(cos1x)=x\cos (\cos^{-1} x) = x\;, 1x1-1 \leq x \leq 1

tan1(tanx)=x\tan^{-1} (\tan x) = x\;, π2xπ2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}

tan(tan1x)=x\tan (\tan^{-1} x) = x\;, -\infty < xx < \infty

Trigonometric Identity:

cos2θ=cos2θsin2θ\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta

  • Introduction
    Application of the Cancellation Laws

    Introduction to Evaluating Inverse Trigonometric Functions


  • 1.
    Understanding the Use of Inverse Trigonometric Functions

    Find the angles for each of the following diagrams.

    a)
    Evaluating inverse trigonometric functions

    b)
    Evaluating inverse trigonometric functions


  • 2.

    Find the angle for the following isosceles triangle.

    Evaluating inverse trigonometric functions

  • 3.
    Determining the Angles in Exact Values by Using Special Triangles

    Find the angles for each of the following diagrams in exact value.

    a)
    Evaluating inverse trigonometric functions

    b)
    Evaluating inverse trigonometric functions


  • 4.
    Application of the Cancellation Laws

    Solve the following inverse trigonometric functions:

    a)
    sin1(sin3π4)\sin^{-1} (\sin \frac{3\pi}{4})


  • 5.
    Solving Expressions With One Inverse Trigonometry

    Solve the following inverse trigonometric functions:

    a)
    cos112\cos^{-1} \frac{1}{2}

    b)
    sin112\sin^{-1} \frac{1}{2}


  • 6.
    Evaluating Expressions With a Combination of Inverse and Non-Inverse Trigonometry

    Solve the following inverse trigonometric functions:

    a)
    sin(cos132)\sin (\cos^{-1} \frac{\sqrt 3}{2})

    b)
    cos(sin123)\cos (\sin^{-1} \frac{2}{3})

    c)
    cos(2tan12)\cos (2\tan^{-1} \sqrt 2)

    d)
    cos(sin1x)\cos (\sin^{-1} x)


  • 7.
    Special Cases: Evaluating Functions With Numbers Outside of the Restrictions

    Solve the following inverse trigonometric functions:

    a)
    cos1(cos3π2)\cos^{-1} (\cos \frac{3\pi}{2})

    b)
    sin1(sin5π2)\sin^{-1} (\sin \frac{5\pi}{2})