14: Appendix
- Page ID
- 1407
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Graphs of the Parent Functions
Graphs of the Trigonometric Functions
Trigonometric Identities
| Pythagorean Identities | \(\begin{align} \cos ^2 t+ \sin ^2 t &=1 \\ 1+ \tan ^2 t &= \sec ^2 t \\ 1+ \cot ^2 t &= \csc ^2 t \end{align}\) |
| Even-Odd Identities | \(\begin{align} \cos(−t) &= \cos t \\ \sec (−t) &= \sec t \\ \sin (−t) &=− \sin t \\ \tan (−t) &=− \tan t \\ \csc (−t) &= − \csc t \\ \cot (−t) &=− \cot t \end{align}\) |
| Cofunction Identities | \(\begin{align} \cos t &= \sin (\frac{π}{2}−t) \\ \sin t &= \cos (\frac{π}{2}−t) \\ \tan t &= \cot (π2−t) \\ \cot t &= \tan (\frac{π}{2}−t) \\ \sec t &= \csc (\frac{π}{2}−t) \\ \csc t &= \sec (\frac{π}{2}−t) \end{align}\) |
| Fundamental Identities | \(\begin{align} \tan t &= \frac{\sin t}{\cos t} \\ \sec t &= \frac{1}{\cos t} \\ \csc t &= \frac{1}{\sin t} \\ \cot t &= \frac{1}{\tan t}=\frac{\cos t}{\sin t} \end{align}\) |
| Sum and Difference Identities | \(\begin{align} \cos (α+β) &= \cos α \cos β −\sin α \sin β \\ \cos (α−β) &= \cos α \cos β+\sin α \sin β \\ \sin (α+β) &= \sin α \cos β+\cos α \sin β \\ \sin (α−β) &= \sin α \cos β−\cos α \sin β \\ \tan (α+β) &= \frac{\tan α+\tan β}{1−\tan α \tan β} \\ \tan (α−β) &= \frac{\tan α− \tan β}{1+\tan α \tan β} \end{align}\) |
| Double-Angle Formulas | \(\begin{align} \sin (2θ) &=2 \sin θ \cos θ \\ \cos (2θ) &= \cos ^2 θ−\sin ^2 θ \\ \cos (2θ) &= 1−2 \sin ^2 θ \\ \cos (2θ) &= 2 \cos ^2 θ−1 \\ \tan (2θ)= \frac{2 \tan θ}{1− \tan ^2 θ} \end{align}\) |
| Half-Angle Formulas | \(\begin{align} \sin \frac{α}{2} &= ±\sqrt{\frac{1−\cos α}{2}} \\ \cos \frac{α}{2} &=±\sqrt{\frac{1+\cos α}{2}} \\ \tan \frac{α}{2} &=± \sqrt{\frac{1− \cos α}{1+ \cos α}} \\ \tan \frac{α}{2} &= \frac{\sin α}{1+ \cos α} \\ \tan \frac{α}{2} &=\frac{1− \cos α}{\sin α} \end{align}\) |
| Reduction Formulas | \(\begin{align} \sin^2 θ &= \frac{1− \cos (2θ)}{2} \\ \cos ^2 θ &= \frac{1+ \cos (2θ)}{2} \\ \tan ^2 θ &= \frac{1− \cos (2θ)}{1+ \cos (2θ)} \end{align}\) |
| Product-to-Sum Formulas | \(\begin{align} \cos α \cos β &=\frac{1}{2}[ \cos(α−β)+\cos(α+β) ] \\ \sin α \cos β &= \frac{1}{2}[ \sin (α+β)+\sin (α−β) ] \\ \sin α \sin β &= \frac{1}{2} [ \cos (α−β)− \cos (α+β) ] \\ \cos α \sin β &=\frac{1}{2}[ \sin (α+β)− \sin (α−β) ] \end{align}\) |
| Sum-to-Product Formulas | \(\begin{align} \sin α+\sin β &= 2 \sin (\frac{α+β}{2}) \cos (\frac{α−β}{2}) \\ \sin α− \sin β &=2 \sin (\frac{α−β}{2}) \cos (\frac{α+β}{2}) \\ \cos α−\cos β &=−2 \sin (\frac{α+β}{2}) \sin (\frac{α−β}{2}) \\ \cos α+ \cos β &=2 \cos (\frac{α+β}{2}) \cos (\frac{α−β}{2}) \end{align}\) |
| Law of Sines | \(\begin{align} \frac{\sin α}{a} &= \frac{\sin β}{b}=\frac{ \sin γ}{c} \\ \frac{a}{\sin α} &= \frac{b}{\sin β} = \frac{c}{\sin γ}\end{align}\) |
| Law of Cosines | \(\begin{align} a^2 &=b^2+c^2−2 bc \cos α \\ b^2 &= a^2+c^2−2ac \cos β \\ c^2 &= a^2+b^2−2ab \cos γ \end{align}\) |