Relations between functions of the same arc

Mathematics

Know the main relationships between functions of the same arc, defined from the sine and cosine functions.

Per Elainy MarcianoPosted in 12/11/2020 - 16:10

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From the functions sine and cosine of the same arc, other trigonometric functions: tangent, secant, cosecant and cotangent.

See below the main relations between functions of the same arc.

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Tangent is defined as the ratio of sine and cosine.

$\dpi{120} \boldsymbol{tan (x) \frac{sin (x)}{cos (x)}}$

As there is no division by zero, only the values of $\dpi{120} x$ for which $\dpi{120} cos (x)\neq 0$ .

Therefore, we must have $\dpi{120} \boldsymbol{x\neq \frac{\pi}{2}+ k \pi, k\in \mathbf{Z}}$ .

The secant is defined as the inverse of the cosine function:

$\dpi{120} \boldsymbol{sec (x) \frac{1}{cos (x)}}$

Being $\dpi{120} \boldsymbol{x\neq \frac{\pi}{2}+ k \pi, k\in \mathbf{Z}}$ .

The cosecant is defined as the inverse of the sine function:

$\dpi{120} \boldsymbol{csc (x) \frac{1}{sin (x)}}$

For what $\dpi{120} sin (x)\neq 0$ , we should have $\dpi{120} \boldsymbol{x\neq k \pi, k\in \mathbf{Z}}$ .

The cotangent is given by the inverse of the tangent function:

$\dpi{120} \boldsymbol{cot (x) \frac{1}{tan (x)} \frac{cos (x)}{sin (x)}}$

Being $\dpi{120} \boldsymbol{x\neq k \pi, k\in \mathbf{Z}}$ .

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