Sine

Table of Contents

Understanding the Sine Ratio

In this chapter we focus on the sine ratio itself: what it measures, how to compute it in right triangles, how it behaves for different angles, and a first connection to the unit circle. General ideas of “trigonometric ratio” and “angle” are assumed from the parent chapters.

Sine in a Right Triangle

Consider a right triangle. Choose one of the non-right angles and call it $\theta$ (theta). Relative to this angle we name the sides:

Hypotenuse: the side opposite the right angle (longest side).
Opposite side: the side directly across from angle $\theta$.
Adjacent side: the side next to angle $\theta$ that is not the hypotenuse.

The sine of the angle $\theta$ is defined (in a right triangle) as:

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

This definition only depends on the angle $\theta$, not on the overall size of the triangle. Any right triangle with the same angle $\theta$ will give the same ratio.

Example: Computing a Sine from Side Lengths

Suppose a right triangle has:

hypotenuse of length $10$,
the side opposite angle $\theta$ has length $6$.

Then
$$
\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{10} = 0.6.
$$

Here you have computed the sine of that angle just from the sides.

Example: Finding a Side from a Sine

If you know $\sin\theta$ and the hypotenuse, you can find the opposite side:

$$
\text{opposite} = (\sin\theta)\cdot\text{hypotenuse}.
$$

If $\sin\theta = 0.5$ and the hypotenuse is $12$, then
$$
\text{opposite} = 0.5 \times 12 = 6.
$$

Using a calculator, you would often start with $\theta$ and get the value of $\sin\theta$.

Basic Properties of the Sine Ratio

Range: How Big or Small Can Sine Be?

In a right triangle, the hypotenuse is always the longest side, so the opposite side is never longer than the hypotenuse. This gives

$$
0 \le \frac{\text{opposite}}{\text{hypotenuse}} \le 1,
$$

so for angles $0^\circ \le \theta \le 90^\circ$:

$$
0 \le \sin\theta \le 1.
$$

Later, when sine is extended to larger angles (and negative angles) via the unit circle, $\sin\theta$ can be negative, but it will always stay between $-1$ and $1$:

$$
-1 \le \sin\theta \le 1.
$$

Special Right Triangles and Exact Sine Values

Some angles have “nice” exact sine values, especially in two common special right triangles.

The $30^\circ$–$60^\circ$–$90^\circ$ Triangle

In this triangle, the side lengths follow the ratio:

$$
\text{short leg} : \text{long leg} : \text{hypotenuse}
= 1 : \sqrt{3} : 2.
$$

If we take the angle opposite the short leg to be $30^\circ$ and the angle opposite the long leg to be $60^\circ$:

For $\theta = 30^\circ$ (opposite side is the short leg):
$$
\sin 30^\circ = \frac{1}{2}.
$$
For $\theta = 60^\circ$ (opposite side is the long leg):
$$
\sin 60^\circ = \frac{\sqrt{3}}{2}.
$$

The $45^\circ$–$45^\circ$–$90^\circ$ Triangle

This is an isosceles right triangle. If each leg has length $1$, the hypotenuse is $\sqrt{2}$. Each acute angle is $45^\circ$ and the side opposite a $45^\circ$ angle has length $1$, so

$$
\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.
$$

(Here we used the common simplification $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.)

Sine of $0^\circ$ and $90^\circ$

You can think about these using either a very “flattened” right triangle or the unit circle idea (discussed later):

At $\theta = 0^\circ$, the opposite side shrinks to $0$, so
$$
\sin 0^\circ = 0.
$$
At $\theta = 90^\circ$, the side opposite the angle becomes the hypotenuse, so
$$
\sin 90^\circ = 1.
$$

Combining with earlier special angles:

$\sin 0^\circ = 0$
$\sin 30^\circ = \dfrac{1}{2}$
$\sin 45^\circ = \dfrac{\sqrt{2}}{2}$
$\sin 60^\circ = \dfrac{\sqrt{3}}{2}$
$\sin 90^\circ = 1$

These values are used frequently in trigonometry.

Using Sine to Solve Right-Triangle Problems

For right-triangle problems that involve an angle and a side, sine is one of the main tools (together with cosine and tangent, covered in their own chapters).

Typical patterns:

Given angle and hypotenuse, find opposite side:
$$
\text{opposite} = \sin\theta \cdot \text{hypotenuse}.
$$
Given opposite side and hypotenuse, find the angle:
Use the inverse sine function (discussed more fully elsewhere):
$$
\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right),
$$
sometimes written as $\arcsin$.

Example: Height of a Tree

You stand $20$ meters from the base of a tree. The angle between the ground and your line of sight to the top of the tree is $35^\circ$. Assume your eyes are at ground level for simplicity.

Let:

$\theta = 35^\circ$,
hypotenuse = line of sight to the top of the tree (unknown),
opposite = height of the tree (what we want),
adjacent = $20$ m.

We know
$$
\sin 35^\circ = \frac{\text{opposite}}{\text{hypotenuse}},
$$
but the hypotenuse is not given. Instead, we should use:

$$
\tan 35^\circ = \frac{\text{opposite}}{\text{adjacent}},
$$

which belongs to the tangent chapter.

However, if the hypotenuse were given (for example, a ladder problem where the ladder length is known), then sine would directly give the height:

Suppose a $10$ m ladder leans against a wall at an angle of $50^\circ$ with the ground. The height it reaches is

$$
\text{height} = \sin 50^\circ \cdot 10.
$$

A calculator will give a numerical answer.

Sine and the Unit Circle (Preview)

Later, sine will be defined using the unit circle. The key idea is:

The unit circle is a circle of radius $1$ centered at the origin.
For an angle $\theta$ drawn from the positive $x$-axis, the point on the unit circle has coordinates $(x, y)$.
Then
$$
\sin\theta = y,
$$
the $y$-coordinate of that point.

This definition:

works for any angle, not just $0^\circ$ to $90^\circ$,
naturally explains why $-1 \le \sin\theta \le 1$,
ties sine to graphs of functions (handled in later chapters).

For now, it is enough to remember that in a right triangle sine is $\dfrac{\text{opposite}}{\text{hypotenuse}}$, and in the unit circle picture sine is the vertical coordinate of a point on the circle.

Sine as a Function (Brief View)

Thinking of sine as a function means:

Input: an angle $\theta$ (in degrees or radians, as specified).
Output: a number between $-1$ and $1$.

We write this as $y = \sin\theta$ or $f(\theta) = \sin\theta$.

In later chapters on trigonometric functions and their graphs, you will see how $\sin\theta$: