Key takeaway: The tangent ratio of an acute angle in a right triangle is the length of the opposite leg divided by the length of the adjacent leg, so it needs no hypotenuse.
Grade 10 – GeometryG-SRT.C.6Trigonometry
Definition of Tangent
The tangent ratio links the two legs of a right triangle. For an acute angle, divide the leg across from it by the leg touching it. Because only legs are involved, it is the go-to ratio when the hypotenuse is unknown or unreachable, as when measuring the height of a tree from a spot on the ground.
Unlike sine and cosine, the tangent ratio is not limited to values below 1. For small angles it is tiny, it equals 1 at \(45^{\circ}\), and it climbs without bound as the angle approaches \(90^{\circ}\). It also matches the slope of a line that rises at that angle.
Formula and notation
Written \(\tan\theta\):
\[ \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \]
Find an angle from two legs with \(\theta = \tan^{-1}\!(\frac{\text{opp}}{\text{adj}})\).
Key facts
- It is the TOA in SOH-CAH-TOA.
- It can be larger than 1, unlike sine and cosine of an acute angle.
- Its value at 45 degrees is exactly 1, since both legs are equal.
- It equals the slope of a line that makes that angle with the horizontal.
Tangent example
Marta stands 40 ft from the base of a tree. Her eyes are 5.5 ft above the ground, and she sights the treetop at \(36^{\circ}\) above horizontal. Use \(\tan 36^{\circ} \approx 0.7265\) to find the tree’s height.
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The leg above eye level is opposite the angle and the 40 ft leg is adjacent, so \(\tan 36^{\circ} = \frac{h}{40}\).
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Solve: \(h = 40 \times 0.7265 \approx 29.1\) ft above her eyes.
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Add her eye height: \(29.1 + 5.5 = 34.6\) ft.
The tree is about 34.6 ft tall.
Common mistake
Watch out
Marta writes her answer as 29.1 ft, the tree’s full height.
Why it is wrong
The triangle starts at her eye level, not at the ground. The tangent ratio gives only the part of the tree above her eyes. Adding the 5.5 ft from the ground to her eyes gives the full height of 34.6 ft.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Trigonometry