Topic 3.2 Notes – Sine, Cosine, and Tangent

Sine, Cosine, and Tangent on the Unit Circle

An angle is in standard position when its vertex is at the origin, its initial ray sits on the positive $x$-axis, and its terminal ray comes from rotating that ray. Counterclockwise rotation gives a positive angle. Clockwise rotation gives a negative angle.

The unit circle is the circle centered at the origin with equation

$$x^2+y^2=1$$

If the terminal ray hits the unit circle at $P=(x,y)$, then

$$P=(\cos\theta,\sin\theta)$$

That order matters. Cosine is the $x$-coordinate. Sine is the $y$-coordinate. In the diagram, the point $\left(\frac12,\frac{\sqrt3}{2}\right)$ shows how those coordinates match $(\cos t,\sin t)$.

Tangent comes from slope. Since the terminal ray goes through $(0,0)$ and $(x,y)$,

$$\tan\theta=\frac{y}{x}=\frac{\sin\theta}{\cos\theta}$$

when $x\neq 0$.

This is the bridge from earlier right-triangle trig. On the unit circle, the triangle ratios turn into coordinate ratios, so trig now works for any angle, not only acute ones.

Standard Position, Radians, and Coterminal Angles

Radians measure angle by arc length. The formula is

$$\theta=\frac{s}{r}$$

where $s$ is arc length and $r$ is radius. On the unit circle, $r=1$, so $\theta=s$. That means radian measure matches directed arc length.

Benchmark angles you should know cold:

• $\frac{\pi}{2}=90^\circ$
• $\pi=180^\circ$
• $\frac{3\pi}{2}=270^\circ$
• $2\pi=360^\circ$

Conversions:

$$\text{radians}=\text{degrees}\left(\frac{\pi}{180}\right) \qquad \text{degrees}=\text{radians}\left(\frac{180}{\pi}\right)$$

Coterminal angles share the same terminal ray, so they differ by full revolutions:

• Radians: $\theta+2\pi k$
• Degrees: $\theta+360k$

where $k$ is any integer.

So $\frac{\pi}{3}$ and $\frac{7\pi}{3}$ are coterminal, and $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$ are coterminal. Same terminal ray means same sine, cosine, and tangent value whenever tangent exists.

Finding Trig Values from a Point or Ray

When the point is not on the unit circle, use the radius first. If $P=(x,y)$, then

$$r=\sqrt{x^2+y^2}$$

and

$$\sin\theta=\frac{y}{r},\qquad \cos\theta=\frac{x}{r},\qquad \tan\theta=\frac{y}{x}$$

if $x\neq 0$.

Dividing by $r$ normalizes the point to the unit circle:

$$\left(\frac{x}{r},\frac{y}{r}\right)$$

Worked example with $P=(-8,15)$:

1. Find the radius
   $$r=\sqrt{(-8)^2+15^2}=\sqrt{64+225}=17$$
2. Use the ratios
   $$\cos\theta=-\frac{8}{17},\qquad \sin\theta=\frac{15}{17},\qquad \tan\theta=-\frac{15}{8}$$

Since the point is in Quadrant II, the signs make sense. Negative $x$, positive $y$.

A common test mistake is treating $(-8,15)$ as $(\cos\theta,\sin\theta)$. You can only do that on the unit circle.

Signs and Axis Values

The signs come from coordinates.

Axis angles are the exact values you should know together. The unit circle below also shows the common benchmark angles, but for this section, focus on the four points on the axes.

• $0 \rightarrow (1,0)$, so $\cos 0=1$, $\sin 0=0$, $\tan 0=0$
• $\frac{\pi}{2} \rightarrow (0,1)$, so cosine $=0$, sine $=1$, tangent undefined
• $\pi \rightarrow (-1,0)$, so cosine $=-1$, sine $=0$, tangent $=0$
• $\frac{3\pi}{2} \rightarrow (0,-1)$, so cosine $=0$, sine $=-1$, tangent undefined

What Students Miss Under Time Pressure

• Cosine is $x$, sine is $y$. The point is $(\cos\theta,\sin\theta)$, not the other way around.
• Tangent is undefined on vertical rays because $x=0$, so you would divide by zero.
• Horizontal rays give tangent $0$ because the slope is $0$.
• Extra revolutions do not change trig values if the angles are coterminal.
• Degree and radian measures get mixed all the time, especially with benchmark angles.