Trigonometric Ratios

In this chapter, we focus on the three basic trigonometric ratios: sine, cosine, and tangent. The later chapters will treat each ratio separately; here we concentrate on what these ratios are as a group, why they are defined the way they are, and how they relate to one another.

Right triangles and “opposite–adjacent–hypotenuse”

Trigonometric ratios for an angle are first defined using right triangles. Consider a right triangle (one angle is $90^\circ$) and focus on one of the acute angles (less than $90^\circ$); call it $\theta$.

Relative to this angle $\theta$:

• The hypotenuse is the side opposite the right angle. It is always the longest side.
• The opposite side is the side directly across from angle $\theta$.
• The adjacent side is the side next to angle $\theta$ that is not the hypotenuse.

These names (opposite, adjacent, hypotenuse) depend on which angle $\theta$ you are using. If you switch to the other acute angle, the opposite and adjacent sides swap roles, but the hypotenuse stays the same.

We will use:

• $h$ for the hypotenuse,
• $o$ for the opposite side (relative to $\theta$),
• $a$ for the adjacent side (relative to $\theta$).

Defining the three primary trigonometric ratios

The three basic trigonometric ratios for an acute angle $\theta$ in a right triangle are defined as fractions of side lengths:

• Sine of $\theta$:
  $$
  \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{o}{h}.
  $$
• Cosine of $\theta$:
  $$
  \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{h}.
  $$
• Tangent of $\theta$:
  $$
  \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{o}{a}.
  $$

These ratios are numbers (often decimals) that measure how the sides of a right triangle compare for a given angle.

For non-right triangles or angles outside $0^\circ$–$90^\circ$, these ratios are generalized using the unit circle, which is treated in a later chapter. Here we stay with right triangles and acute angles.

A common memory aid for these definitions is the pattern:

• $\sin\theta$: Sine = Opposite / Hypotenuse
• $\cos\theta$: Cosine = Adjacent / Hypotenuse
• $\tan\theta$: Tangent = Opposite / Adjacent

This gives the well-known mnemonic “SOH–CAH–TOA”:

• $\text{SOH}: \sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$,
• $\text{CAH}: \cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$,
• $\text{TOA}: \tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$.

Why these ratios do not depend on triangle size

A crucial idea is that $\sin\theta$, $\cos\theta$, and $\tan\theta$ depend only on $\theta$, not on the overall size of the triangle.

If you draw two right triangles with the same acute angle $\theta$ but different sizes (one is a scaled-up version of the other), then:

• All corresponding sides are in proportion.
• So the ratios $o/h$, $a/h$, and $o/a$ are the same in both triangles.

Thus:

• $\sin\theta$, $\cos\theta$, and $\tan\theta$ are functions of the angle only.
• Once $\theta$ is fixed, every right triangle with that angle has the same trigonometric ratios.

This is what allows us to tabulate or compute values like $\sin 30^\circ$ or $\cos 45^\circ$ independent of any particular triangle.

Basic relationships between the trigonometric ratios

Even before studying each ratio in detail, some fundamental relationships connect them.

The Pythagorean identity for sine and cosine

By the Pythagorean theorem in a right triangle:
$$
a^2 + o^2 = h^2.
$$

Divide both sides by $h^2$:
$$
\left(\frac{a}{h}\right)^2 + \left(\frac{o}{h}\right)^2 = 1.
$$

Using the definitions:

• $\dfrac{a}{h} = \cos\theta$,
• $\dfrac{o}{h} = \sin\theta$,

we get the key identity:
$$
\sin^2\theta + \cos^2\theta = 1.
$$

Here $\sin^2\theta$ is shorthand for $(\sin\theta)^2$, and similarly for $\cos^2\theta$.

This equation expresses a fundamental link between sine and cosine.

Tangent as a ratio of sine and cosine

Using the definitions:
$$
\sin\theta = \frac{o}{h},\quad \cos\theta = \frac{a}{h},\quad \tan\theta = \frac{o}{a}.
$$

Compute $\dfrac{\sin\theta}{\cos\theta}$:
$$
\frac{\sin\theta}{\cos\theta} = \frac{\dfrac{o}{h}}{\dfrac{a}{h}}
= \frac{o}{h} \cdot \frac{h}{a}
= \frac{o}{a}
= \tan\theta.
$$

So we have:
$$
\tan\theta = \frac{\sin\theta}{\cos\theta},
$$
whenever $\cos\theta \neq 0$.

This relationship shows tangent is not independent; it is built from sine and cosine.

Using trigonometric ratios to find unknown sides

In a right triangle, if you know:

• one acute angle $\theta$ (other than $90^\circ$), and
• the length of one side,

then trigonometric ratios allow you to find the other sides.

The general strategy is:

1. Identify which sides are opposite, adjacent, and hypotenuse relative to angle $\theta$.
2. Choose the ratio (sine, cosine, or tangent) that involves:
• the side you know,
• and the side you want to find.
3. Set up an equation using the definition of the chosen ratio.
4. Solve the equation for the unknown side.

For instance:

• If you know $\theta$ and the hypotenuse $h$, and you want the opposite side $o$:
  $$
  \sin\theta = \frac{o}{h} \quad\Rightarrow\quad o = h\sin\theta.
  $$
• If you know $\theta$ and adjacent side $a$, and you want hypotenuse $h$:
  $$
  \cos\theta = \frac{a}{h} \quad\Rightarrow\quad h = \frac{a}{\cos\theta}.
  $$
• If you know $\theta$ and adjacent side $a$, and you want opposite side $o$:
  $$
  \tan\theta = \frac{o}{a} \quad\Rightarrow\quad o = a\tan\theta.
  $$

The actual numerical values $\sin\theta$, $\cos\theta$, and $\tan\theta$ for specific angles (like $30^\circ$, $45^\circ$, and $60^\circ$) will be studied in more detail elsewhere, including exact values and calculator use.

Using trigonometric ratios to find unknown angles

If you know two sides of a right triangle, you can use trigonometric ratios to find one of the acute angles.

For example, suppose you know:

• opposite side $o$,
• hypotenuse $h$.

From the definition:
$$
\sin\theta = \frac{o}{h}.
$$

To solve for $\theta$, you use the inverse trigonometric function (read “arc-sine”):
$$
\theta = \arcsin\left(\frac{o}{h}\right).
$$

Similarly:

• If you know $a$ (adjacent) and $h$ (hypotenuse):
  $$
  \cos\theta = \frac{a}{h} \quad\Rightarrow\quad \theta = \arccos\left(\frac{a}{h}\right).
  $$
• If you know $o$ (opposite) and $a$ (adjacent):
  $$
  \tan\theta = \frac{o}{a} \quad\Rightarrow\quad \theta = \arctan\left(\frac{o}{a}\right).
  $$

These inverse functions and their properties are discussed in more depth when angles and trigonometric functions are extended beyond simple right-triangle settings.

Domains and reasonable values of the ratios (right-triangle view)

In the right-triangle context (with angle $\theta$ between $0^\circ$ and $90^\circ$):

• All side lengths are positive, so:
• $\sin\theta > 0$,
• $\cos\theta > 0$,
• $\tan\theta > 0$.
• The hypotenuse is the longest side, so:
  $$
  0 < \frac{o}{h} < 1,\quad 0 < \frac{a}{h} < 1.
  $$
  Therefore:
  $$
  0 < \sin\theta < 1,\quad 0 < \cos\theta < 1.
  $$
• The ratio $\tan\theta = \dfrac{o}{a}$ can be any positive real number, depending on the relative sizes of $o$ and $a$:
• If $o < a$, then $0 < \tan\theta < 1$.
• If $o > a$, then $\tan\theta > 1$.

The extension of these ratios to angles beyond $0^\circ$–$90^\circ$ (including negative angles and radians) is handled using the unit circle in a later chapter.

Summary of the three primary trigonometric ratios

For an acute angle $\theta$ in a right triangle:

• $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$,
• $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$,
• $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

They satisfy:

• $\sin^2\theta + \cos^2\theta = 1$,
• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ (when $\cos\theta \neq 0$).

These ratios allow you to move between angles and side lengths in right triangles. Later chapters will:

• examine each ratio (sine, cosine, tangent) more deeply,
• introduce additional trigonometric ratios (like cotangent, secant, and cosecant),
• and extend these ideas beyond right triangles using the unit circle and trigonometric functions.