# Precalculus- Concepts Through Functions

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## 5.3 Properties of the Trigonometric Functions

 ${\displaystyle \sin \theta =y}$ ${\displaystyle \cos \theta =x}$ ${\displaystyle \tan \theta ={\frac {y}{x}},x\neq 0}$ ${\displaystyle \csc \theta ={\frac {1}{y}},y\neq 0}$ ${\displaystyle \sec \theta ={\frac {1}{x}},x\neq 0}$ ${\displaystyle \cot \theta ={\frac {x}{y}},y\neq 0}$

### 5.3.1 The Domain and Range of Trigonometric Functions

 The domain of the sine function is the set of all real numbers. The domain of the cosine function is the set of all real numbers.

For the secant and tangent functions, the x-coordinate of P cannot be 0 since this results in a division by 0. On the unit circle, ${\displaystyle x=0}$ on two such points, (0,1) and (0,-1). These points correspond to the angles ${\displaystyle {\frac {\pi }{2}}}$ and ${\displaystyle {\frac {3\pi }{2}}}$ and then any odd integer multiples thereof. Such angles must be excluded from the domain of the tangent and secant functions.

 The domain of the secant function is the set of all real numbers, except odd integer multiples of ${\displaystyle {\frac {\pi }{2}}}$. The domain of the tangent function is the set of all real numbers, except odd integer multiples of ${\displaystyle {\frac {\pi }{2}}}$.

For the cosecant and cotangent functions, the y-coordinate of P cannot be 0 since this results in a division by 0. On the unit circle, ${\displaystyle y=0}$ at two such points, (1,0) and (-1,0). These points correspond to the angles 0 and ${\displaystyle \pi }$ and then any integer multiples thereof. Such angles must be excluded from the domain of the cosecant and cotangent functions.

 The domain of the cosecant function is the set of all real numbers, except integer multiples of ${\displaystyle \pi }$ The domain of the cotangent function is the set of all real numbers, except integer multiples of ${\displaystyle \pi }$

If ${\displaystyle -1\leq x\leq 1}$ and ${\displaystyle -1\leq y\leq 1}$ and ${\displaystyle \sin \theta =y}$ and ${\displaystyle \cos \theta =x}$, then

 ${\displaystyle -1\leq \sin \theta \leq 1}$ and ${\displaystyle -1\leq \cos \theta \leq 1}$

The domain of arcsin (x) is the range of sin (x) , which is [−1, 1] . The range of arcsin (x) is the domain of sin (x) , i.e. [− π /2 , π /2 ].

The domain of arcos(x) is −1 ≤ x ≤ 1 , the range of arcos(x) is [0 , π] , arcos(x) is the angle in [0, π] whose cosine is x.

The domain of arctan(x) is all real numbers , the range of arctan is from −π/2 to π/2 radians inclusive .