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Section 7.1 Trigonometric Identities Summary Sheet
Below is a reference sheet of all trigonometric identities in this chapter.
Definition 7.1.1. Definitions for tangent, cotangent, secant, and cosecant.
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\(\displaystyle \tan{(x)} = \frac{\sin(x)}{\cos(x)}\)
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\(\displaystyle \cot{(x)} = \frac{\cos(x)}{\sin(x)}\)
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\(\displaystyle \sec{(x)} = \frac{1}{\cos(x)}\)
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\(\displaystyle \csc{(x)} = \frac{1}{\sin(x)}\)
Definition 7.1.2. Odd and even trigonometric functions.
Sine, tangent, cotangent, and cosecant are odd functions, while cosine and secant are even functions.
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\(\displaystyle \sin(-x) = -\sin(x)\)
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\(\displaystyle \tan(-x) = -\tan(x)\)
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\(\displaystyle \cot(-x) = -\cot(x)\)
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\(\displaystyle \csc(-x) = -\csc(x)\)
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\(\displaystyle \cos(-x) = \cos(x)\)
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\(\displaystyle \sec(-x) = \sec(x)\)
Definition 7.1.3. Pythagorean Identities.
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\(\displaystyle \sin^2(x) + \cos^2(x) = 1\)
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\(\displaystyle \tan^2(x) + 1 = \sec^2(x)\)
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\(\displaystyle \cot^2(x) + 1 = \csc^2(x)\)
Definition 7.1.4. Sum and Difference Identities.
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\(\displaystyle \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)\)
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\(\displaystyle \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)\)
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\(\displaystyle \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)\)
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\(\displaystyle \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)\)
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\(\displaystyle \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}\)
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\(\displaystyle \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\)
Definition 7.1.5. Double-angle and half-angle identities.
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\begin{align*}
\cos(2\theta) \amp = \cos^2(\theta) - \sin^2(\theta)\\
\amp = 1 - 2\sin^2(\theta)\\
\amp = 2\cos^2(\theta) - 1
\end{align*}
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\(\displaystyle \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}\)
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\(\displaystyle \sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
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\(\displaystyle \sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}\)
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\(\displaystyle \cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}\)
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\(\displaystyle \tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\)
Definition 7.1.6. Product-to-sum and Sum-to-product identities.
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\(\displaystyle \cos(\alpha)\cos(\beta) = \frac12 \left[\cos(\alpha - \beta) + \cos(\alpha+\beta)\right]\)
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\(\displaystyle \sin(\alpha)\cos(\beta) = \frac12 \left[\sin(\alpha + \beta) + \sin(\alpha+\beta)\right]\)
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\(\displaystyle \sin(\alpha)\sin(\beta) = \frac12 \left[\cos(\alpha - \beta) - \cos(\alpha+\beta)\right]\)
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\(\displaystyle \cos(\alpha)\sin(\beta) = \frac12 \left[\sin(\alpha + \beta) - \sin(\alpha-\beta)\right]\)
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\(\displaystyle \sin(\alpha) + \sin(\beta) = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\)
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\(\displaystyle \sin(\alpha) - \sin(\beta) = 2\sin\left(\frac{\alpha - \beta}{2}\right)\cos\left(\frac{\alpha+\beta}{2}\right)\)
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\(\displaystyle \cos(\alpha) - \cos(\beta) = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\)
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\(\displaystyle \cos(\alpha) + \cos(\beta) = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\)
