Trigonometry

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Content you are assumed to know (from Methods)

• Angles in degrees in radians (and converting between units)
• Defining sin and cos using the unit circle
• Graphs of y = sin x, y = cos x, and y = tan x
• The Pythagorean Identity
  • $si{n}^2(x)+co{s}^2(x)=1$
• Symmetry properties of trig functions
  • $sin(\theta )=sin(\pi -\theta )$
• Exact values of trig functions for integer multiples of $\frac{\pi }{2},\frac{\pi }{4}$ and $\frac{\pi }{6}$ radians

Basic Trig Functions

$sin(-\theta )=-sin(\theta )$ $cos(-\theta )=cos(\theta )$ $tan(-\theta )=-tan(\theta )$ $sin(\theta +\frac{\pi }{2})=cos(\theta )$ $cos(\theta -\frac{\pi }{2})=sin(\theta )$

General solutions to trig equations

• Remember that a trig equation such as $cos(\theta )=0.3$ has infinitely many solutions
  • There are 2 solutions even in one revolution of the unit circle
  • However, a scientific calculator, if asked to evaluate $co{s}^{-1}(0.3)$ will only give one value
  • How does it decide which value to give?
• Inverse trig functions
  • By convention: $co{s}^{-1}$ has range $\{\theta |0\le \theta \le \pi \}$
  • By convention: $si{n}^{-1}$ has range $\{\theta |-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}\}$
  • By convention: $ta{n}^{-1}$ has range $\{\theta |-\frac{\pi }{2}<\theta <\frac{\pi }{2}\}$

Sums of Trig Identities

$a\sin (x)+b\cos (x))$ $=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\sin (x)+\frac{b}{\sqrt{a^2+b^2}}\cos (x)\right)$ $=\sqrt{a^2+b^2}\left(\cos (\alpha )\sin (x)+\sin (\alpha )\cos (x)\right)$ $=\sqrt{a^2+b^2}\sin (x+\alpha )$

Reciprocal Trigonometric Functions

$\frac{1}{\cos \theta }=\sec \theta$

$\frac{1}{\sin \theta }=\text{cosec}\theta$

• The functions above are basically asymptotes of their respective denominators, look at images online

$\frac{1}{\tan \theta }=\cot \theta =\frac{\cos \theta }{\sin \theta }$

• Basically just a tan graph reflected over the y axis

Pythagorean Identities

${\cos }^2\theta +{\sin }^2\theta =1$