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Section A.2 Trigonometric Identities
Identity A.2.1 . Pythagorean Identities.
\(\displaystyle \sin ^2(\theta)+\cos^2(\theta)=1\)
\(\displaystyle \tan ^2(\theta)+1 = \sec^2(\theta)\)
\(\displaystyle 1+\cot^2(\theta)=\csc^2(\theta)\)
Identity A.2.2 . Sum and Difference Formulas.
Identity A.2.3 . Double Angle Formulas.
\(\displaystyle \sin(2\theta)=2\sin\theta\cos\theta\)
\(\displaystyle \cos(2\theta)=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1\)
\(\displaystyle \tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}\)
Identity A.2.4 . Power Reduction Formulas.
\(\displaystyle \sin^2\theta=\frac{1-\cos2\theta}{2}\)
\(\displaystyle \cos^2\theta=\frac{1+\cos2\theta}{2}\)
\(\displaystyle \tan^2\theta=\frac{1-\cos2\theta}{1+\cos2\theta}\)
Identity A.2.5 . Half Angle Formulas.
\(\sin\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{2}}\) or \(\sin\frac{\theta}{2}=-\sqrt{\frac{1-\cos\theta}{2}}\text{,}\) depending on which quadrant \(\theta\) is in.
\(\cos\frac{\theta}{2}=\sqrt{\frac{1+\cos\theta}{2}}\) or \(\cos\frac{\theta}{2}=-\sqrt{\frac{1+\cos\theta}{2}}\text{,}\) depending on which quadrant \(\theta\) is in.
\(\tan\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}\) or \(\tan\frac{\theta}{2}=-\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}\text{,}\) depending on which quadrant \(\theta\) is in.
