Sine, Cosine, and Tangent

You Already Know Two of Them

From the unit circle: any point P at angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$.

• The x-coordinate is cosine
• The y-coordinate is sine

That’s the definition. But where does it come from?

The Right Triangle Connection

Drop a line from P to the x-axis. You get a right triangle:

• Hypotenuse = 1 (the radius)
• Adjacent side (along x-axis) = $\cos \theta$
• Opposite side (vertical) = $\sin \theta$

From any right triangle, not just the unit circle:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

On the unit circle the hypotenuse is 1, so these simplify to just “opposite” and “adjacent”.

Tangent

The third trig function. It’s defined as:

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

Or from the triangle:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

What Does Tangent Mean Geometrically?

$\sin \theta$ over $\cos \theta$ is the same as $y$ over $x$, which is rise over run.

Tangent is the slope of the terminal side.

But there’s more. Draw a vertical line at the right edge of the unit circle, at $x = 1$. This line is tangent to the circle. Extend the terminal side until it hits this line.

The length of that segment on the vertical line is $\tan \theta$. That’s where the name comes from.

When Tangent Breaks

At $\theta = 90°$, the terminal side points straight up. It never hits the vertical tangent line, they’re parallel.

Algebraically: $\cos 90° = 0$, and $\tan \theta = \sin \theta / \cos \theta$. Dividing by zero is undefined.

The same happens at $270°$, and at any angle where the terminal side is vertical.

The Three Functions Together

The Pythagorean Identity

Every point on the unit circle satisfies $x^2 + y^2 = 1$.

Since $x = \cos \theta$ and $y = \sin \theta$:

$\cos^2 \theta + \sin^2 \theta = 1$

This is always true, for any angle. It connects sine and cosine and will show up everywhere.