Cosine Definition
Description
The Cosine function (abbreviated as $\cos$) is one of the fundamental trigonometric ratios. Together with Sine and Tangent, it relates the angles of a right-angled triangle to the lengths of its sides. Specifically, cosine connects the angle to the adjacent side and the hypotenuse.
The mnemonic **SOH CAH TOA** is the standard way to remember this: * Sine = Opposite / Hypotenuse * **C**osine = **A**djacent / **H**ypotenuse * Tangent = Opposite / Adjacent
On the Unit Circle (a circle with radius 1 centered at the origin), the cosine of an angle $\theta$ is defined as the **x-coordinate** of the point where the terminal side of the angle intersects the circle. This definition allows cosine to be extended to any real number, including negative angles and angles larger than 360°, where it forms a periodic wave pattern essential for physics and engineering.
History & Origins
The history of cosine parallels that of sine. Ancient India (c. 500 AD): While sine (jya) was the primary focus, the concept of "complementary sine" or koti-jya was used to describe what we now call cosine. It literally meant the sine of the complementary angle (90° - $\theta$). Edmund Gunter (1620): He introduced the term "co-sine" as an abbreviation for complementi sinus (sine of the complement), solidifying the relationship $\cos(\theta) = \sin(90^\circ - \theta)$. Euler (1700s): Leonhard Euler popularized the modern notation $\cos$ and treated it as a function of a real number rather than just a geometric line segment, paving the way for modern analysis.
Unit Circle Definition
The cosine function is best understood geometrically on the unit circle.
Draw a circle with radius $r=1$ on a Cartesian coordinate system.
Draw an angle $\theta$ starting from the positive x-axis.
The terminal side of the angle touches the circle at a point $P(x, y)$.
Drop a perpendicular line from $P$ to the x-axis to form a right triangle.
The **hypotenuse** is the radius ($r=1$).
The **adjacent** side is the horizontal distance from the origin, which is $x$.
By definition, $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{x}{1} = x$.
Therefore, on the unit circle, cosine is simply the x-coordinate.
Variables
| Symbol | Meaning |
|---|---|
θ | Angle (degrees or radians) |
adjacent | Side next to the angle (touching it) |
hypotenuse | Longest side (opposite the 90° angle) |
Examples
Basic Calculation
Problem: Find cos(60°)
Solution:
Ramp Construction
Problem: You are building a ramp with a 10-meter long plank. If the ramp must make a 30° angle with the ground, how far from the base of the wall will it start?
Solution: ~8.66 meters
- Identify knowns: Hypotenuse (plank) = 10m, Angle = 30°.
- Identify unknown: Adjacent side (distance on ground).
- Choose ratio: CAH (Cosine = Adjacent / Hypotenuse).
- Equation: $\cos(30^\circ) = \frac{x}{10}$.
- Solve: $x = 10 \times \cos(30^\circ)$.
- Calculate: $x = 10 \times 0.866 = 8.66$ meters.
Vector Components
Problem: A force vector of 50 Newtons is applied at a 45° angle to the horizontal. What is the horizontal component ($F_x$)?
Solution: ~35.35 N
- Formula: $F_x = F \times \cos(\theta)$.
- Substitute: $F_x = 50 \times \cos(45^\circ)$.
- Know that $\cos(45^\circ) \approx 0.707$.
- Calculate: $50 \times 0.707 = 35.35$ N.
Common Mistakes
Confusing Adjacent and Opposite
The Adjacent side ALWAYS touches the angle you are interested in (and the right angle). The Opposite side does not touch the angle.
Calculator in Wrong Mode
$\\cos(90^\circ) = 0$, but $\\cos(90 \\text{ rad}) \\approx -0.448$. Always check if you are in Degrees (DEG) or Radians (RAD).
Real-World Applications
Computer Graphics (Lighting)
In 3D rendering, the "Lambert's Cosine Law" calculates light intensity. The brightness of a surface depends on the cosine of the angle between the light source and the surface normal. If light hits straight on (0°), it's brightest ($\cos(0)=1$).
Physics: Work
The formula for Work is $W = F d \cos(\theta)$. Only the force component acting *in the direction* of movement does work. Cosine filters out the wasted force pushing sideways.
Frequently Asked Questions
Why is cos(0) = 1?
At 0 degrees, the angle is flat. The "adjacent" side lies perfectly along the hypotenuse, so their lengths are equal. Ratio = 1/1 = 1.
What is the "Law of Cosines"?
It is a generalized Pythagorean Theorem for NON-right triangles: $c^2 = a^2 + b^2 - 2ab \cos(C)$. It works for any triangle.