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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

4d3e042a38f6b45a487871f8b3e98d4c.png4ff6073e65b92eef216921d212d131f8.png

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

2b56c3f89f685c430c6cc22358093016.png

Negative angles

Periodicity Identities

Cofunction Identities

\(\sin\left(-\theta \right)=-\sin\theta\)

\(\sin\left(\theta ±{360}°\right)=\sin\theta\)

\(\sin\left({90}°-\theta \right)=\cos\theta\)

\(\cos\left(-\theta \right)=\cos\theta\)

\(\cos\left(\theta ±{360}°\right)=\cos\theta\)

\(\cos\left({90}°-\theta \right)=\sin\theta\)

\(\tan\left(-\theta \right)=-\tan\theta\)

\(\tan\left(\theta ±{180}°\right)=\tan\theta\)

\(\sin\left({90}°+\theta \right)=\cos\theta\)

\(\cos\left({90}°+\theta \right)=- \sin\theta\)

bfb7596890eaba3fbefbb9bf7d2e0dfc.png

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

\(\sin\left(\theta +\beta\right)=\sin\theta\cos \beta +\cos\theta\sin \beta\)

\(\sin\left(2\theta \right)=2\sin\theta\cos \theta\)

\(\sin\left(\theta -\beta \right)=\sin\theta\cos \beta -\cos\theta\sin \beta\)

\(\cos\left(2\theta \right)={\cos}^{2}\theta -{\sin}^{2}\theta\)

\(\cos\left(\theta +\beta \right)=\cos\theta\cos \beta -\sin\theta\sin \beta\)

\(\cos\left(2\theta \right)=1-2{\sin}^{2}\theta\)

\(\cos\left(\theta -\beta \right)=\cos\theta\cos \beta +\sin\theta\sin \beta\)

\(\cos\left(2\theta \right)=2{\cos}^{2}\theta - 1\)

\(\)

\(\tan\left(2\theta \right)=\frac{ \sin 2 \theta }{ \cos 2 \theta }\)