Tangent (Trigonometric Ratio)
In short: The tangent ratio compares the two legs of a right triangle, the one facing a chosen corner over the one touching it, and works as a direct gauge of steepness.
GeometryTrigonometryG-SRT.C.6
What is Tangent?
The tangent ratio turns a tilt into a rise-over-run number. For a corner \(A\), \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\), so the hypotenuse plays no part: only the two legs matter.
Unlike sine and cosine it is not capped at 1. It is small for gentle tilts, equals 1 at \(45^{\circ}\), and grows without bound as the tilt nears \(90^{\circ}\). A straight line that leans \(\theta\) above the horizontal has slope \(m = \tan\theta\), which links the idea to graphing.
How to spot it
- A story gives the horizontal distance to a tall object plus the angle of sight, and asks for its height.
- Both legs are known and the tilt between the hypotenuse and a leg is wanted.
- The slanted side is neither given nor needed, which points to the tangent ratio instead of sine or cosine.
- A roof pitch, ramp or hillside is described by its rise and its run.
The method
- To use the tangent ratio, label the legs from the marked corner: opposite faces it, adjacent touches it.
- Write \(\tan\theta = \frac{\text{opp}}{\text{adj}}\) and plug in the known numbers.
- For the opposite leg, multiply: \(\text{opp} = \text{adj} \times \tan\theta\). For the corner, use \(\tan^{-1}\).
- Round last, and compare with the picture: a steep tilt must give a tangent above 1.
Tangent: solved example
Problem
A surveyor stands 45 feet from the foot of an oak and measures \(38^{\circ}\) up to the top branch. Ignoring eye height, how tall is the oak? Use \(\tan 38^{\circ} \approx 0.7813\).
Solution
The 45 feet runs along the ground beside the corner, so it is the adjacent leg, and the oak is the opposite leg.
From \(\tan 38^{\circ} = \frac{h}{45}\): \(h = 45 \times 0.7813 \approx 35.2\). The oak is about 35.2 feet tall.
Try it
Question 1
A triangular brace has legs of 9 and 12 inches. If corner \(P\) faces the 9, find \(\tan P\).
Show the answer
\(\tan P = \frac{9}{12} = 0.75\).
Question 2
A corner has \(\tan\theta = 2.4\) and the leg touching it is 5. How long is the leg facing it?
Show the answer
\(5 \times 2.4 = 12\).
Tangent vs. Tangent to a Circle
One word, two ideas. Here it is a number: a quotient of two legs belonging to a right triangle. The circle version is a line that brushes a circle at exactly one point and stands perpendicular to the radius there. You compute one with a calculator; you draw the other with a ruler.
Keep going
Lessons
Worksheets and quizzes
Related words
Whole dictionary: A–Z · Geometry · Trigonometry