Sine Definition
Description
The Sine function (abbreviated as $\sin$) is one of the three primary trigonometric ratios, forming the foundation of trigonometry along with Cosine and Tangent. It relates the angle of a right-angled triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse.
The mnemonic **SOH CAH TOA** helps remember this definition: * **S**ine = **O**pposite / **H**ypotenuse * Cosine = Adjacent / Hypotenuse * Tangent = Opposite / Adjacent
Beyond triangles, the sine function is fundamental in describing periodic phenomena. On the unit circle (a circle with radius 1), if you measure an angle $\theta$ from the positive x-axis, the y-coordinate of the point on the circle is exactly $\sin(\theta)$. This allows sine to be defined for all real numbers, not just angles between 0° and 90°.
History & Origins
The concept of sine has a rich history spanning across civilizations. Ancient India (c. 500 AD): The mathematician Aryabhata used the term ardha-jya (half-chord) to describe sine. This was later shortened to jya or jiva. Islamic Golden Age (c. 800 AD): Arab mathematicians translated the Sanskrit texts. They transliterated jiva into Arabic as jiba. Since Arabic is written without short vowels, it was written as jb. Later readers interpreted this as jayb, which means "pocket" or "fold" in Arabic. Medieval Europe (c. 1150 AD): When Gherard of Cremona translated these Arabic texts into Latin, he translated jayb into the Latin word for pocket/fold: sinus. This mistranslation stuck, giving us the modern word "sine".
Unit Circle Definition
The sine function extends beyond right triangles using the unit circle.
Draw a circle with radius $r=1$ centered at $(0,0)$.
Draw a line from the origin making an angle $\theta$ with the positive x-axis.
This line intersects the circle at a point $P(x,y)$.
Draw a vertical line from $P$ down to the x-axis to form a right triangle.
The hypotenuse is the radius, so $H = 1$.
The side opposite to $\theta$ is the vertical height $y$.
By definition, $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{y}{1} = y$.
Thus, on the unit circle, sine is simply the y-coordinate.
Variables
| Symbol | Meaning |
|---|---|
θ | Angle (in degrees or radians) |
opposite | Length of side opposite to the angle |
hypotenuse | Length of the longest side (opposite 90°) |
Examples
Basic Calculation
Problem: Find sin(30°)
Solution:
Ladder Problem
Problem: A 10-meter ladder leans against a wall making a 60° angle with the ground. How high up the wall does it reach?
Solution: ~8.66 meters
- Identify knowns: Hypotenuse (ladder) = 10m, Angle $\theta = 60^\circ$.
- Identify unknown: Opposite side (height of wall).
- Choose ratio: SOH (Sine = Opposite / Hypotenuse).
- Equation: $\sin(60^\circ) = \frac{h}{10}$.
- Solve for h: $h = 10 \times \sin(60^\circ)$.
- Calculate: $h = 10 \times 0.866 = 8.66$ meters.
Sound Waves
Problem: A pure tone is modeled by $y(t) = A \sin(2\pi f t)$. If amplitude A=5 and frequency f=440Hz, what is the value at t=0.001s?
Solution: ~1.91
- Equation: $y = 5 \sin(2\pi \times 440 \times 0.001)$.
- Calculate angle: $2\pi \times 0.44 \approx 2.76$ radians.
- Compute sine: $\sin(2.76) \approx 0.38$.
- Multiply by Amplitude: $5 \times 0.38 = 1.9$.
Common Mistakes
Degrees vs Radians
Calculators have two modes. $\sin(30^\circ) = 0.5$, but $\sin(30 \text{ rad}) = -0.98$. Always check your mode!
Confusing Sine and Cosine
Remember SOH CAH TOA. Sine is Opposite, Cosine is Adjacent. If the side touches the angle, it's Adjacent (Cosine).
Real-World Applications
Sound and Music
Musical tones are sound waves composed of sine waves at different frequencies. Synthesizers create sounds by adding multiple sine waves together.
Alternating Current (AC)
The electricity in your wall outlet alternates direction in a smooth sine wave pattern. Engineers use sine functions to model voltage and current over time.
Frequently Asked Questions
Why is sine between -1 and 1?
In a right triangle, the opposite side can never be longer than the hypotenuse, so the ratio cannot exceed 1. On the unit circle, the y-coordinate goes from -1 (bottom) to 1 (top).
What is inverse sine?
Inverse sine ($\sin^{-1}$ or arcsin) does the reverse: it takes a ratio and tells you the angle. If $\sin(\theta) = 0.5$, then $\theta = 30^\circ$.