Precalculus Review

Trigonometric functions

Sine, cosine and tangent functions. Given a right triangle with acute angle $\theta$ , the sine, the cosine and the tangent functions of the radian $\theta$ are respectively given by

$\displaystyle \sin\theta = \dfrac{ b }{c}; \hspace{0.2in} \cos\theta = \dfrac{ a }{c}; \hspace{0.2in} \tan\theta = \dfrac{ b }{a}.$

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Now suppose that the hypotenuse of right triangle is . Then (i) the opposite side equals $\sin\theta$ and (ii) the adjacent side equals $\cos\theta$ . By Pythagorean theorem we obtain

$\displaystyle \sin^2\theta + \cos^2\theta = 1$

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We can also observe that

$\displaystyle \tan\theta = \dfrac{\sin\theta}{\cos\theta}.$

Cosecant, secant and cotangent functions. We can define the cosecant, the secant and the cotangent functions of $\theta$ as the respective reciprocal of the sine, the cosine and the tangent functions of $\theta$ .

$\displaystyle \csc\theta = \dfrac{1}{\sin\theta}; \hspace{0.2in} \sec\theta = \dfrac{1}{\cos\theta}; \hspace{0.2in} \cot\theta = \dfrac{1}{\tan\theta}.$

All the six functions introduced above are collectively called the trigonometric functions.