# The Cosine and Sine Curves: $$y=cos(x)$$ & $$y = sin(x)$$

## (How to draw them and what their properties are)

The cosine and sine curves, written: $y = cos(x)\quad \text{and} \quad y = sin(x)$ Show us how the value of

• $$y=cos(x)$$ shows us the horizontal coordinate

## The Cosine Curve, $$y = cos(x)$$

Remembering the definition of cosine, with a unit circle, the cosine curve $$y=cos(x)$$ corresponds to the horizontal coordinate of a point moving around a unit circle.

Looking at the cosine curve $$y=cos(x)$$ we can see quite clearly that

• it oscillates between its minimum value $$-1$$ and maximum value $$1$$. This confirms the fact that the range of $$cos(x)$$ is: $-1 \leq cos(x) \leq 1$
• it is periodic, with period $$T = 360^{\circ}$$.
Indeed no matter which point we look at on the curve: its
• It reaches its maximum value $$cos(x) = 1$$ when $$x=0^{\circ}, 360^{\circ},720^{\circ}, \dots$$.
• It equals to zero $$cos(x) = 0$$ and cuts the $$x$$-axis when $$x=90^{\circ}, 270^{\circ},450^{\circ}, \dots$$ (all $$x$$-intercepts are $$180^{\circ}$$ apart from each other).

## The Sine Curve, $$y = sin(x)$$

Remembering the definition of sine, with a unit circle, the sine curve $$y=sin(x)$$ corresponds to the vertical coordinate of a point moving around a unit circle.

Looking at the sine curve $$y=sin(x)$$ we can see quite clearly that

• just like the cosine curve: it oscillates between its minimum value $$-1$$ and maximum value $$1$$. This confirms the fact that the range of $$sin(x)$$ is: $-1 \leq sin(x) \leq 1$
• it is periodic, with period $$T = 360^{\circ}$$.
Indeed no matter which point we look at on the curve: its
• It reaches its maximum value $$cos(x) = 1$$ when $$x=90^{\circ}, 450^{\circ},810^{\circ}, \dots$$.
• It equals to zero $$sin(x) = 0$$ and cuts the $$x$$-axis when $$x=0^{\circ}, 180^{\circ},360^{\circ}, \dots$$ (all $$x$$-intercepts are $$180^{\circ}$$ apart from each other).

## Method: Sketching $$y=cos(x)$$ & $$y=sin(x)$$

To sketch both $$y=cos(x)$$ and $$y=sin(x)$$ the idea is to plot:

• points at which the curve reaches the maximum height of $$1$$. For $$y=sin(x)$$: that'
• points at which the curve reaches the minimum height of $$-1$$,
• points at which the curve cuts the $$x$$-axis.
Once this is done, we draw a smooth curve passing through these points.

This method is best explained in the tutorial below.