Evaluating inverse trigonometric functions

Evaluating inverse trigonometric functions

Lessons

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

  • 1.
    Introduction to Evaluating Inverse Trigonometric Functions

  • 2.
    Application of the Cancellation Laws

    Solve the following inverse trigonometric functions:

    a)
    sin(sin10.5)\sin (\sin^{-1} 0.5)

    b)
    cos1(cosπ4)\cos^{-1} (\cos \frac{\pi}{4})

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


  • 3.
    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}


  • 4.
    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)


  • 5.
    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})