TBIL-Precal Trigonometric Identities

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Appendix B Trigonometric Identities

Identity B.0.1. Pythagorean Identities.

$\sin^{2} (θ) + \cos^{2} (θ) = 1$
$\tan^{2} (θ) + 1 = \sec^{2} (θ)$
$1 + \cot^{2} (θ) = \csc^{2} (θ)$

Identity B.0.2. Sum and Difference Formulas.

$\sin (α + β) = \sin α \cos β + \cos α \sin β$
$\sin (α - β) = \sin α \cos β - \cos α \sin β$
$\cos (α + β) = \cos α \cos β - \sin α \sin β$
$\cos (α - β) = \cos α \cos β + \sin α \sin β$
$\tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β}$
$\tan (α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β}$

Identity B.0.3. Double Angle Formulas.

$\sin (2 θ) = 2 \sin θ \cos θ$
$\cos (2 θ) = \cos^{2} θ - \sin^{2} θ = 1 - 2 \sin^{2} θ = 2 \cos^{2} θ - 1$
$\tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ}$

Identity B.0.4. Power Reduction Formulas.

$\sin^{2} θ = \frac{1 - \cos 2 θ}{2}$
$\cos^{2} θ = \frac{1 + \cos 2 θ}{2}$
$\tan^{2} θ = \frac{1 - \cos 2 θ}{1 + \cos 2 θ}$

Identity B.0.5. Half Angle Formulas.

$\sin \frac{θ}{2} = \sqrt{\frac{1 - \cos θ}{2}}$ or $\sin \frac{θ}{2} = - \sqrt{\frac{1 - \cos θ}{2}},$ depending on which quadrant $θ$ is in.
$\cos \frac{θ}{2} = \sqrt{\frac{1 + \cos θ}{2}}$ or $\cos \frac{θ}{2} = - \sqrt{\frac{1 + \cos θ}{2}},$ depending on which quadrant $θ$ is in.
$\tan \frac{θ}{2} = \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ}$ or $\tan \frac{θ}{2} = - \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ},$ depending on which quadrant $θ$ is in.