Mathematics » Functions II » The Cosine Function

# Revision of The Cosine Function

## Functions of the form $$y = \cos \theta$$ for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$

• The period is $$\text{360}\text{°}$$ and the amplitude is $$\text{1}$$.

• Domain: $$[\text{0}\text{°};\text{360}\text{°}]$$

For $$y = \cos \theta$$, the domain is $$\{ \theta: \theta \in \mathbb{R} \}$$, however in this case, the domain has been restricted to the interval $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$.

• Range: $$[-1;1]$$

• $$x$$-intercepts: $$(\text{90}\text{°};0)$$, $$(\text{270}\text{°};0)$$

• $$y$$-intercept: $$(\text{0}\text{°};1)$$

• Maximum turning points: $$(\text{0}\text{°};1)$$, $$(\text{360}\text{°};1)$$

• Minimum turning point: $$(\text{180}\text{°};-1)$$

## Functions of the form $$y = a \cos \theta + q$$

Cosine functions of the general form $$y = a \cos \theta + q$$, where $$a$$ and $$q$$ are constants.

The effects of $$a$$ and $$q$$ on $$f(\theta) = a \cos \theta + q$$:

• The effect of $$q$$ on vertical shift

• For $$q>0$$, $$f(\theta)$$ is shifted vertically upwards by $$q$$ units.

• For $$q<0$$, $$f(\theta)$$ is shifted vertically downwards by $$q$$ units.

• The effect of $$a$$ on shape

• For $$a>1$$, the amplitude of $$f(\theta)$$ increases.

• For $$0<a<1$$, the amplitude of $$f(\theta)$$ decreases.

• For $$a<0$$, there is a reflection about the $$x$$-axis.

• For $$-1 < a < 0$$, there is a reflection about the $$x$$-axis and the amplitude decreases.

• For $$a < -1$$, there is a reflection about the $$x$$-axis and the amplitude increases.