Tangent Definition
Description
The Tangent function (abbreviated as $\tan$) is one of the three fundamental trigonometric ratios, alongside Sine and Cosine. While Sine and Cosine relate an angle to the hypotenuse, the Tangent relates the two legs of a right-angled triangle to each other: the side **Opposite** the angle and the side **Adjacent** to the angle.
The mnemonic **TOA** (part of SOH CAH TOA) helps remember this definition: * **T**angent = **O**pposite / **A**djacent
**Geometric Interpretation:** * **Slope:** In the Cartesian coordinate system, the tangent of an angle $\theta$ is exactly the **slope** (rise over run) of the line forming that angle with the positive x-axis. $\tan(\theta) = \frac{\Delta y}{\Delta x} = \frac{\sin(\theta)}{\cos(\theta)}$. * **The "Touching" Line:** On the Unit Circle, if you draw a vertical line tangent to the circle at $x=1$, the tangent of $\theta$ is the length of the segment on that vertical line from the x-axis to the extension of the angle's terminal side. This is why it's called "tangent" (touching).
**Key Properties:** * **Range:** Unlike Sine and Cosine which are trapped between -1 and 1, Tangent can take any real value from $-\infty$ to $+\infty$. * **Asymptotes:** The function is undefined at $90^\circ$ ($\\frac{\pi}{2}$) and $270^\circ$ ($3\\frac{\pi}{2}$), because the Adjacent side (cosine) becomes zero, leading to division by zero. On the graph, these appear as vertical asymptotes.
History & Origins
The concept of the tangent is actually older than sine and cosine in terms of practical use, largely due to the study of shadows. Ancient Shadows (Gnomonics): Ancient civilizations used gnomons (vertical sticks) to tell time. The relationship between the height of the stick and the length of its shadow is essentially the tangent (or cotangent) function. Thales of Miletus famously used this principle to measure the height of the Great Pyramid by waiting until the length of his own shadow equaled his height (when $\tan(\theta) = 1$, angle = 45°). Islamic Mathematics (c. 800-900 AD): The Persian mathematician Al-Marwazi produced the first table of tangents (shadow lengths). Later, Al-Biruni and Al-Battani defined tangent as a trigonometric function distinct from shadow tables. Thomas Fincke (1583): The Danish mathematician who first coined the term "tangent" in his book Geometria Rotundi. It comes from the Latin tangere, meaning "to touch," referring to the geometric interpretation on the unit circle.
Derivation from Sine and Cosine
We can derive the tangent formula directly from the definitions of sine and cosine.
Recall Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$.
Recall Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.
Divide Sine by Cosine: $\frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\text{Opposite}}{\text{Hypotenuse}}}{\frac{\text{Adjacent}}{\text{Hypotenuse}}}$.
Simplify the fraction: The "Hypotenuse" terms cancel out.
Result: $\frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{Opposite}}{\text{Adjacent}}$.
By definition, this is $\tan(\theta)$.
Geometric Proof (Unit Circle)
Why is it the length of the tangent line?
Draw a unit circle (radius $r=1$) and an angle $\theta$ at the origin.
Draw a vertical line touching the circle at $(1,0)$. This is the tangent line.
Extend the terminal side of the angle until it hits this vertical line at point $T(1, y)$.
Form a right triangle with the origin $(0,0)$, the point $(1,0)$, and $T(1,y)$.
The Adjacent side is the radius, which is 1.
The Opposite side is the height $y$.
Calculate ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{y}{1} = y$.
Thus, the y-coordinate of the intersection on the tangent line is the tangent of the angle.
Variables
| Symbol | Meaning |
|---|---|
θ | Angle (degrees or radians) |
opposite | Side opposite to the angle |
adjacent | Side next to the angle (not hypotenuse) |
Examples
Basic Calculation
Problem: Find tan(45°)
Solution:
Measuring Building Height
Problem: You are standing 50 meters from the base of a building. You measure the angle of elevation to the top as 60°. How tall is the building?
Solution: ~86.6 meters
- Identify knowns: Adjacent = 50m, Angle = 60°.
- Identify unknown: Opposite (Height).
- Choose ratio: TOA (Tangent = Opposite / Adjacent).
- Equation: $\tan(60^\circ) = \frac{h}{50}$.
- Know that $\tan(60^\circ) = \sqrt{3} \approx 1.732$.
- Solve: $h = 50 \times 1.732$.
- Result: $h \approx 86.6$ meters.
Slope of a Roof
Problem: A roof rises 4 meters for every 12 meters of horizontal run. What is the angle of the roof?
Solution: ~18.4°
- Identify: Opposite (Rise) = 4, Adjacent (Run) = 12.
- Formula: $\tan(\theta) = \frac{4}{12} = 0.333$.
- Use Inverse Tangent: $\theta = \tan^{-1}(0.333)$.
- Calculate: $\theta \approx 18.43^\circ$.
Common Mistakes
Using Hypotenuse
Tangent NEVER uses the hypotenuse. It is only Opposite and Adjacent. If you have the hypotenuse, use Sine or Cosine.
tan(90°)
Students often try to calculate $\tan(90^\circ)$ and get "Error" or assume it is 0. It is UNDEFINED (infinity) because the adjacent side becomes 0.
Real-World Applications
Surveying & Map Making
Surveyors use an instrument called a Theodolite to measure horizontal and vertical angles. By knowing the angle and a baseline distance (adjacent), they use the tangent function to calculate elevations of mountains or heights of landmarks without climbing them.
Physics: Coefficient of Friction
If you place a block on a ramp and slowly tilt it, the block will start sliding at a specific angle $\theta$. The "coefficient of static friction" $\mu$ is exactly equal to $\tan(\theta)$. This simple experiment allows physicists to determine friction properties of materials.
Frequently Asked Questions
Why is tan(45°) = 1?
At 45 degrees, the triangle is an isosceles right triangle. The Opposite and Adjacent sides are exactly the same length. Any number divided by itself is 1.
How is Tangent related to Slope?
They are the same thing! The slope $m$ of a line is defined as $\frac{\text{Rise}}{\text{Run}}$, which is exactly $\frac{\text{Opposite}}{\text{Adjacent}}$, or $\tan(\theta)$.