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## 4.6 Summary (EMCGP)

 Pythagorean Identities Ratio Identities $${\cos}^{2}\theta +{\sin}^{2}\theta =1$$ $$\tan\theta =\frac{\sin\theta }{\cos\theta }$$ $${\cos}^{2}\theta = 1 - {\sin}^{2}\theta$$ $$\frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}$$ $${\sin}^{2}\theta = 1 - {\cos}^{2}\theta$$

Special angle triangles

 θ $$\text{0}$$° $$\text{30}$$° $$\text{45}$$° $$\text{60}$$° $$\text{90}$$° $$\cos θ$$ $$\text{1}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{\sqrt{2}}$$ $$\frac{1}{2}$$ $$\text{0}$$ $$\sin θ$$ $$\text{0}$$ $$\frac{1}{2}$$ $$\frac{1}{\sqrt{2}}$$ $$\frac{\sqrt{3}}{2}$$ $$\text{1}$$ $$\tan θ$$ $$\text{0}$$ $$\frac{1}{\sqrt{3}}$$ $$\text{1}$$ $$\sqrt{3}$$ undef

CAST diagram and reduction formulae

 Negative angles Periodicity Identities Cofunction Identities $$\sin\left(-\theta \right)=-\sin\theta$$ $$\sin\left(\theta ±{360}°\right)=\sin\theta$$ $$\sin\left({90}°-\theta \right)=\cos\theta$$ $$\cos\left(-\theta \right)=\cos\theta$$ $$\cos\left(\theta ±{360}°\right)=\cos\theta$$ $$\cos\left({90}°-\theta \right)=\sin\theta$$ $$\tan\left(-\theta \right)=-\tan\theta$$ $$\tan\left(\theta ±{180}°\right)=\tan\theta$$ $$\sin\left({90}°+\theta \right)=\cos\theta$$ $$\cos\left({90}°+\theta \right)=- \sin\theta$$
 Area Rule Sine Rule Cosine Rule $$\text{Area}=\frac{1}{2}bc\sin \hat{A}$$ $$\frac{\sin \hat{A}}{a}=\frac{\sin \hat{B}}{b}=\frac{\sin \hat{C}}{c}$$ $${a}^{2}={b}^{2}+{c}^{2}-2bc\cos \hat{A}$$ $$\text{Area}=\frac{1}{2}ab\sin \hat{C}$$ $$a \sin \hat{B} = b \sin \hat{A}$$ $${b}^{2}={a}^{2}+{c}^{2}-2ac\cos \hat{B}$$ $$\text{Area}=\frac{1}{2}ac\sin \hat{B}$$ $$b \sin{C} = c \sin \hat{B}$$ $${c}^{2}={a}^{2}+{b}^{2}-2ab\cos \hat{C}$$ $$a \sin{C} = c \sin \hat{A}$$
 Compound Angle Identities Double Angle Identities $$\sin\left(\theta +\beta\right)=\sin\theta\cos \beta +\cos\theta\sin \beta$$ $$\sin\left(2\theta \right)=2\sin\theta\cos \theta$$ $$\sin\left(\theta -\beta \right)=\sin\theta\cos \beta -\cos\theta\sin \beta$$ $$\cos\left(2\theta \right)={\cos}^{2}\theta -{\sin}^{2}\theta$$ $$\cos\left(\theta +\beta \right)=\cos\theta\cos \beta -\sin\theta\sin \beta$$ $$\cos\left(2\theta \right)=1-2{\sin}^{2}\theta$$ $$\cos\left(\theta -\beta \right)=\cos\theta\cos \beta +\sin\theta\sin \beta$$ $$\cos\left(2\theta \right)=2{\cos}^{2}\theta - 1$$  $$\tan\left(2\theta \right)=\frac{ \sin 2 \theta }{ \cos 2 \theta }$$