Cosine

Understanding Cosine

Cosine is one of the three primary trigonometric ratios, alongside sine and tangent. In this chapter, we focus specifically on what cosine measures, how to compute it in right triangles and using the unit circle, and how to interpret its sign and typical values.

Because this chapter is part of “Trigonometric Ratios,” you can assume you already know what an angle is and have seen the general idea of a trigonometric ratio.

Cosine in a Right Triangle

In a right triangle, each acute angle “sees” two sides:

• The hypotenuse (the side opposite the right angle)
• The adjacent side (the side next to the angle, but not the hypotenuse)
• The opposite side (the side across from the angle)

For an acute angle $\theta$ in a right triangle, the cosine of $\theta$ is defined as:
$$
\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
$$

This definition only applies when $\theta$ is an acute angle in a right triangle (that is, $0^\circ < \theta < 90^\circ$), but later, the unit-circle view lets us extend cosine to any angle.

Simple examples

1. Suppose a right triangle has:
• Adjacent side to angle $\theta$: $4$
• Hypotenuse: $5$

Then
$$
\cos\theta = \frac{4}{5} = 0.8.
$$

2. In a $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle where the hypotenuse is $2$ and the shorter leg is $1$:
• For the $60^\circ$ angle, the adjacent side is $\sqrt{3}$ and the hypotenuse is $2$,
  so
  $$
  \cos 60^\circ = \frac{1}{2},\quad \cos 30^\circ = \frac{\sqrt{3}}{2}.
  $$

(You do not need to memorize special-triangle facts here if they are treated elsewhere, but they are common examples.)

Cosine and the Unit Circle

The right-triangle definition works nicely for acute angles, but not for angles larger than $90^\circ$ or for negative angles. To handle all real angles, cosine is defined using the unit circle.

On the unit circle (a circle of radius $1$ centered at the origin $(0,0)$ in the coordinate plane), every angle $\theta$ corresponds to a point with coordinates:
$$
(\cos\theta,\ \sin\theta).
$$

So, for any real angle $\theta$ (measured in degrees or radians, but consistently):

• $\cos\theta$ is the $x$-coordinate of the point on the unit circle at angle $\theta$.

This extension agrees with the right-triangle definition where both apply, but now works for all angles, including obtuse, reflex, and negative angles.

Cosine as horizontal coordinate

If you imagine standing at the origin and looking at a point on the unit circle at angle $\theta$:

• Draw the radius (a line) from the origin to that point.
• Drop a perpendicular from that point down (or up) to the $x$-axis.

The $x$-coordinate of the point is $\cos\theta$. In terms of geometry, cosine tells you:

• how far left or right along the $x$-axis the point lies.

Range and Sign of Cosine

Because points on the unit circle always lie at distance $1$ from the origin:

• The $x$-coordinate of any such point is between $-1$ and $1$.

Therefore, for all real $\theta$:
$$
-1 \le \cos\theta \le 1.
$$

Cosine can be positive, zero, or negative depending on the quadrant of the angle.

• Quadrant I ($0^\circ < \theta < 90^\circ$ or $0 < \theta < \frac{\pi}{2}$):
  $x$-coordinates are positive, so $\cos\theta > 0$.
• Quadrant II ($90^\circ < \theta < 180^\circ$ or $\frac{\pi}{2} < \theta < \pi$):
  $x$-coordinates are negative, so $\cos\theta < 0$.
• Quadrant III ($180^\circ < \theta < 270^\circ$ or $\pi < \theta < \frac{3\pi}{2}$):
  $x$-coordinates are negative, so $\cos\theta < 0$.
• Quadrant IV ($270^\circ < \theta < 360^\circ$ or $\frac{3\pi}{2} < \theta < 2\pi$):
  $x$-coordinates are positive, so $\cos\theta > 0$.

On the axes:

• $\cos 0^\circ = 1$, $\cos 90^\circ = 0$,
• $\cos 180^\circ = -1$, $\cos 270^\circ = 0$, $\cos 360^\circ = 1$.

Cosine of Common Angles

You will often encounter a small set of “standard” angles. In degrees and radians:

• $0^\circ$ ($0$ radians): $\cos 0^\circ = 1$
• $30^\circ$ ($\frac{\pi}{6}$): $\cos 30^\circ = \frac{\sqrt{3}}{2}$
• $45^\circ$ ($\frac{\pi}{4}$): $\cos 45^\circ = \frac{\sqrt{2}}{2}$
• $60^\circ$ ($\frac{\pi}{3}$): $\cos 60^\circ = \frac{1}{2}$
• $90^\circ$ ($\frac{\pi}{2}$): $\cos 90^\circ = 0$

And using symmetry on the unit circle, you can deduce values for angles like $120^\circ$, $135^\circ$, $150^\circ$, etc., often by remembering the reference angle and applying a sign.

Geometric and Practical Meaning of Cosine

Cosine often appears when you measure horizontal components or projections.

Cosine as a projection

If you have a segment of length $r$ making an angle $\theta$ with the positive $x$-axis, the horizontal component (its “shadow” on the $x$-axis) has length:
$$
r\cos\theta.
$$

This is why, for example:

• If a ramp of length $5$ meters rises at an angle of $30^\circ$, the horizontal distance covered is
  $$
  5 \cos 30^\circ.
  $$
• If a force of magnitude $F$ is applied at angle $\theta$ from the horizontal, the horizontal component of the force is $F\cos\theta$.

Cosine is therefore strongly connected to projection and components in geometry and physics.

Cosine of Negative Angles

Using the unit circle, a negative angle $-\theta$ is measured clockwise from the positive $x$-axis. The $x$-coordinates of the points at $\theta$ and $-\theta$ are the same, because the points are reflections across the $x$-axis.

This means:
$$
\cos(-\theta) = \cos\theta
$$
for all real $\theta$. A function with this property is called an even function, but the general study of even/odd functions belongs elsewhere.

Relationship Between Cosine and Sine (Briefly)

Without going into full trigonometric identities, there is one basic relationship between cosine and sine that explains many properties of cosine:

For any real angle $\theta$,
$$
\cos^2\theta + \sin^2\theta = 1.
$$

Here $\cos^2\theta$ means $(\cos\theta)^2$. This comes directly from the equation of the unit circle, $x^2 + y^2 = 1$, where $x = \cos\theta$ and $y = \sin\theta$.

This identity is very useful for:

• checking whether a cosine value is reasonable,
• finding $\cos\theta$ if you know $\sin\theta$, and vice versa (up to a sign, determined by the quadrant).

Using Cosine in Right-Triangle Problems

In right-triangle problems, cosine is especially useful when you know:

• An angle $\theta$ (other than the right angle) and the hypotenuse, and you want the adjacent side, or
• The adjacent side and the hypotenuse, and you want the angle.

Using the definition:
$$
\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}},
$$
you can:

• Solve for the adjacent side:
  $$
  \text{adjacent} = (\text{hypotenuse})\cdot\cos\theta.
  $$
• Or solve for the angle (using a calculator’s inverse cosine function, often written $\cos^{-1}$ or arccos):
  $$
  \theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right).
  $$

How to use calculators or inverse trigonometric functions in detail is addressed in other chapters; here, the key point is that cosine links an angle to the ratio of adjacent side over hypotenuse.

Summary

• In a right triangle, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$.
• On the unit circle, $\cos\theta$ is the $x$-coordinate of the point corresponding to angle $\theta$.
• Cosine values always lie between $-1$ and $1$.
• Cosine describes horizontal components or projections: $r\cos\theta$ is the horizontal component of a length $r$ at angle $\theta$.
• Cosine is an even function: $\cos(-\theta) = \cos\theta$.
• Cosine and sine are connected by $\cos^2\theta + \sin^2\theta = 1$.

These properties make cosine a central tool in geometry, trigonometry, and many applied problems involving angles, directions, and components.