Math 140

Class 16

Inverse Cosine

  • Sends points from the \(x\)-axis to the upper semicircle.
  • Domain: \([-1,1]\) on the \(x\)-axis.
  • Range: \([0, \pi]\).
  • Decreasing.
  • \(\arccos(-1) = \pi\).
  • \(\arccos(1) = 0\).

Inverse Sine

  • Sends points from the \(y\)-axis to the right semicircle.
  • Domain: \([-1,1]\) on the \(y\)-axis.
  • Range: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).
  • Increasing.
  • \(\arcsin(-1) = -\frac{\pi}{2}\).
  • \(\arcsin(1) = \frac{\pi}{2}\).
  • Odd.

Inverse Tangent

  • Sends points from the \(s\)-axis to the right semicircle.
  • Domain: \((-\infty,\infty)\) (the whole \(s\)-axis).
  • Range: \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
  • Increasing.
  • Horizontal asymptote on the left at \(t = -\frac{\pi}{2}\).
  • Horizontal asymptote on the right at \(t = \frac{\pi}{2}\).
  • Odd.

Inverse Tangent

Inverse Cotangent

Inverse Cotangent

  • Sends points from the \(r\)-axis to the upper semicircle.
  • Domain: \((-\infty,\infty)\) (the whole \(r\)-axis).
  • Range: \(\left(0, \pi\right)\).
  • Decreasing.
  • Horizontal asymptote on the left at \(t = \pi\).
  • Horizontal asymptote on the right at \(t = 0\).

Inverse Secant and Cosecant

  • No simple obvious way to restrict the domain
  • There are different ways it can be done
  • “Trigonometry friendly” version
    • \(\operatorname{arcsec}(x) = \arccos\left(1/x\right)\)
    • \(\operatorname{arccsc}(y) = \arcsin\left(1/y\right)\)
  • “Calculus friendly” version
  • You need to figure out from context which one is used.

Inverse Secant

Inverse Secant

x t 1 − 1 π 2 π 3 π 2 arcsec x Commonpart Trigonometry v ersion Calculusv ersion

Inverse Cosecant

Inverse Cosecant

y t 1 − 1 − π 2 π 2 π 3 π 2 arccsc y Commonpart Trigonometry v ersion Calculusv ersion

Example

\(\displaystyle \cos\left(\arcsin \frac{5}{7}\right)\)

Example

\(\displaystyle \sin\left(\arctan 2\right)\)

Example

\(\displaystyle \sin\left(\arccos x\right)\)

Example

\(\displaystyle \sec\left(\operatorname{arccot} r\right)\)

Example

\(\displaystyle \sin\left(2\arcsin y\right)\)

Example

\(\displaystyle \cos\left(2\operatorname{arccsc} y\right)\)

(trigonometry version)

Example

\(\displaystyle \sec\left(2\arctan s\right)\)

Example

\(\displaystyle \cos\left(\arctan \frac{s}{2}\right)\)

Example

\(\displaystyle \cos\left(\frac{\arctan s}{2}\right)\)