The Tangent Ratio: Unlocking the Slopes of Triangles
From Right Angles to Ratios: The Core Definition
Let's start with the basics. A right-angled triangle has one angle measuring exactly $90^\circ$ (the right angle). The side opposite this right angle is the longest side, called the hypotenuse. For any of the other two acute angles (angles less than $90^\circ$), we label the sides relative to that angle.
$ \tan(\theta) = \frac{\text{Length of Opposite Side}}{\text{Length of Adjacent Side}} $
This is the tangent ratio. The word tangent comes from the Latin tangere, meaning "to touch," which relates to its geometric origins.
Consider a right-angled triangle with an angle marked $\theta$.
• The Opposite side is the one directly across from the angle $\theta$.
• The Adjacent side is the one next to the angle $\theta$ that is not the hypotenuse.
The hypotenuse is not used in the tangent ratio directly.
SOH CAH TOA: The Memory Key
To remember the three main trigonometric ratios, students use the mnemonic SOH CAH TOA.$^{[1]}$ Each part represents one ratio:
| Mnemonic Part | Ratio | Formula | Description |
|---|---|---|---|
| SOH | Sine | $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ | Uses Opposite and Hypotenuse. |
| CAH | Cosine | $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ | Uses Adjacent and Hypotenuse. |
| TOA | Tangent | $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ | Uses Opposite and Adjacent, ignoring the Hypotenuse. |
Example 1: In triangle ABC, right-angled at B, angle A = $37^\circ$, and side BC (opposite to A) is 3 cm, side AB (adjacent to A) is 4 cm. What is $\tan(37^\circ)$ for this triangle?
Applying the formula: $\tan(A) = \frac{BC}{AB} = \frac{3}{4} = 0.75$. This means for this specific triangle, the ratio of the side opposite angle A to the side adjacent to it is 0.75.
Using Tangent to Find Missing Sides
The true power of the tangent ratio is in solving for unknown lengths. If you know an acute angle and one side, you can find the other side.
Example 2: Finding the Opposite Side
A kite is flying at the end of a $50$-meter string. The string makes an angle of $63^\circ$ with the ground. How high is the kite above the ground?
Here, the string is the hypotenuse. The height (h) is opposite the $63^\circ$ angle. The distance along the ground is adjacent. We use tangent because we want the opposite side and we likely know the adjacent side (or vice versa). For simplicity, let's assume we know the adjacent side is $d$. The formula is $\tan(63^\circ) = \frac{h}{d}$. If, for instance, $d = 25$ m, then $h = d \times \tan(63^\circ)$. Using a calculator, $\tan(63^\circ) \approx 1.9626$, so $h \approx 25 \times 1.9626 \approx 49.07$ meters.
Example 3: Finding the Adjacent Side
From the top of a $30$-meter lighthouse, the angle of depression$^{[2]}$ to a boat is $10^\circ$. How far is the boat from the base of the lighthouse?
The height ($30$ m) is opposite the angle of elevation from the boat. The distance (d) from the base is adjacent. We have $\tan(10^\circ) = \frac{30}{d}$. Rearranging: $d = \frac{30}{\tan(10^\circ)}$. $\tan(10^\circ) \approx 0.1763$, so $d \approx \frac{30}{0.1763} \approx 170.2$ meters.
Using Tangent to Find Missing Angles (Inverse Tangent)
What if you know the sides but need the angle? This is where the inverse tangent function, written as $\tan^{-1}$ or arctan, comes in.
Example 4: A ramp is $5$ meters long horizontally and rises $1$ meter vertically. What is the angle of the ramp?
Here, opposite = $1$ m, adjacent = $5$ m. So, $\tan(\theta) = \frac{1}{5} = 0.2$.
To find $\theta$, we calculate: $\theta = \tan^{-1}(0.2)$. Using a calculator, $\theta \approx 11.31^\circ$.
Tangent in the Real World: Slope and Steepness
Perhaps the most intuitive application of tangent is in measuring slope. In geometry, geography, and engineering, slope is defined as "rise over run." This is exactly the tangent ratio!
$\text{Slope (m)} = \frac{\text{Rise (Vertical Change)}}{\text{Run (Horizontal Change)}} = \tan(\theta)$
where $\theta$ is the angle the incline makes with the horizontal.
| Scenario | Rise (Opposite) | Run (Adjacent) | Tangent (Slope) | Angle $\theta$ |
|---|---|---|---|---|
| Gentle Hill | $10$ m | $100$ m | $\frac{10}{100}=0.1$ | $\tan^{-1}(0.1) \approx 5.71^\circ$ |
| Steep Ramp | $3$ m | $4$ m | $\frac{3}{4}=0.75$ | $\tan^{-1}(0.75) \approx 36.87^\circ$ |
| Roof Pitch (7:12) | $7$ in | $12$ in | $\frac{7}{12} \approx 0.5833$ | $\tan^{-1}(0.5833) \approx 30.26^\circ$ |
Important Questions
Absolutely. The tangent ratio is simply a fraction (opposite/adjacent). If the opposite side is longer than the adjacent side, the ratio is greater than 1. For example, if opposite = 10 and adjacent = 5, then $\tan(\theta)=2$. This corresponds to a steep angle (about $63.4^\circ$).
In a right-angled triangle where the two acute angles are both $45^\circ$, the opposite and adjacent sides are equal (it's an isosceles right triangle). Therefore, $\tan(45^\circ) = \frac{\text{Equal}}{\text{Equal}} = 1$. This is a fundamental and memorable value in trigonometry.
In the context of right-angled triangles, tangent is defined for all acute angles (from $0^\circ$ to $90^\circ$, not including the endpoints). As the angle approaches $90^\circ$, the opposite side gets very long and the adjacent side gets very short, making the ratio extremely large, tending toward infinity. In broader trigonometry, we say $\tan(90^\circ)$ is undefined.
Connecting Tangent to the Unit Circle
For high school students, the concept extends beyond triangles. On the unit circle (a circle with radius 1 centered at the origin), the tangent of an angle $\theta$ is defined as the y-coordinate divided by the x-coordinate of the point where the terminal side of the angle intersects the circle. This is equivalent to the slope of the line from the origin to that point.
$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
This identity shows that tangent is the ratio of the sine and cosine of the same angle, which aligns perfectly with our triangle definition: $\frac{\text{Opposite/Hypotenuse}}{\text{Adjacent/Hypotenuse}} = \frac{\text{Opposite}}{\text{Adjacent}}$.
Footnote
$[1]$ SOH CAH TOA: A mnemonic acronym used to remember the definitions of the basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
$[2]$ Angle of Depression: The angle formed by a horizontal line (of sight) and the line of sight to an object located below the horizontal line. It is equal in measure to the angle of elevation from the lower object to the observer due to alternate interior angles.