Compositions of Trigonometric and Inverse Trigonometric Functions
- All of these functions are projections from one axis to another.
- Two kinds of compositions:
- “Nonsensical”:
- for example \(\sin(cos(t))\)
- for example \(\arctan(\sin(t))\)
- “Natural”:
- A sequence of projections
- for example \(\sin(\arctan(s))\): \(\displaystyle s \xrightarrow{\arctan} t \xrightarrow{\sin} y\)
Some relations
- \(t = \arccos x \Leftrightarrow x = \cos t\)
- \(t = \arcsin y \Leftrightarrow y = \sin t\)
- \(t = \arctan s \Leftrightarrow s = \tan t = \frac{y}{x}\)
- \(t = \operatorname{arccot} r \Leftrightarrow r = \cot t = \frac{x}{y}\)
- \(t = \operatorname{arcsec} x' \Leftrightarrow x' = \sec t = \frac{1}{\cos t} = \frac{1}{x}\)
- \(t = \operatorname{arccsc} y' \Leftrightarrow y' = \csc t = \frac{1}{\sin t} = \frac{1}{y}\)
- \(\cos^2 t + \sin^2 t = x^2 + y^2 = 1\)
- \(1 + \tan^2 t = \sec^2 t\) or \(1 + s^2 = \frac{1}{x^2}\)
- \(\cot^2 t + 1 = \csc^2 t\) or \(r^2 + 1 = \frac{1}{y^2}\)
Example
\(\sin(\operatorname{arcsec} x)\)
\(\displaystyle \left(\frac{1}{x}\right)^2 + y^2 = 1\)
Example
\(\sec(\operatorname{arctan} s)\)
- \(\displaystyle t = \arctan s\)
- \(s = \tan t = \frac{y}{x}\)
- \(\displaystyle x^2 + y^2 = 1\)
Trigonometric Equations
- \(\cos t = \frac{1}{2}\)
- \(\sin t = -\frac{\sqrt{3}}{2}\)
- \(\sin t = -1\)
- \(\tan t = 1\)
- \(\sec t = -2\)
\(\displaystyle \cos 3x = \frac{\sqrt{2}}{2}\)
\(\displaystyle \sqrt{3}\tan 16x = 1\)
\(\displaystyle \sec^2 \frac{t}{9} = 2\)
\(\displaystyle 9\sin^2 \frac{t}{2} = 4\)
\(\displaystyle \cos(4x) = \cos(3x)\)
\(\displaystyle \cos(4x) - \cos(3x) = 0\)
Sum to Product: \(\cos(u) - \cos(v) = -2\sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
\(\displaystyle \sin(2x) = \cos(x)\)
