Tangent (tan) is the trigonometric ratio of the side opposite an acute angle to the side adjacent to it in a right triangle: tan θ = oppositeadjacent. For legs 3 and 4, tan θ = 34 = 0.75. It is the TOA in SOH CAH TOA, and it equals sin θcos θ.
Opposite and adjacent are named from the angle you are using. Switch to the other acute angle and they trade places, so its tangent is 43.
Formal definition and notation
For an acute angleθ in a right triangle, tan θ is the length of the leg opposite θ divided by the length of the leg adjacent to θ. Because all right triangles with angle θ are similar, the ratio depends only on θ. On the unit circle, tan θ = yx = sin θcos θ for any angle with cos θ ≠ 0, which equals the slope of the terminal side; tangent is undefined at 90° and 270° and repeats every 180°.
Symbol
Read it as
Meaning
tan θ
tangent of theta
opposite over adjacent
tan−1x
inverse tangent of x, or arctan x
the angle whose tangent is x; used to find an angle from two legs
Key facts
Tangent is the only basic trig ratio that does not use the hypotenuse, so it is the one to pick when you know or need the two legs.
From a point 50 m from the base of a tree, the angle of elevation to the top is 35°. How tall is the tree, to the nearest tenth of a meter?
The height is opposite the 35° angle and the 50 m distance is adjacent.
tan 35° = h50.
h = 50 × tan 35° ≈ 50 × 0.7002.
h ≈ 35.0 m.
Answer: The tree is about 35.0 m tall.
Example 3: From tangent to sine and cosine
Test-style
For an acute angle θ, tan θ = 512. Find sin θ and cos θ.
Draw a right triangle with opposite 5 and adjacent 12.
Hypotenuse: √52 + 122 = √169 = 13.
sin θ = 513 and cos θ = 1213.
Check: 513 ÷ 1213 = 512.
Answer:sin θ = 513 and cos θ = 1213.
Common mistakes
✗ Mistaketan θ = adjacentopposite: for legs 3 (opposite) and 4 (adjacent), 43.
✓ Insteadtan θ = 34.
TOA: Tangent = Opposite over Adjacent. 43 is the tangent of the other acute angle.
✗ MistakeUsing the hypotenuse in a tangent: 3 over 5.
✓ Instead35 is sin θ. Tangent uses the two legs: 34.
Sine and cosine use the hypotenuse; tangent never does.
✗ MistakeGetting tan 35° ≈ 0.47 on a calculator.
✓ InsteadIn degree mode, tan 35° ≈ 0.700.
0.47 is the tangent of 35 radians. Check that the calculator is in degree mode.
Not to be confused with
Tangentvs.Tangent line
In circle geometry, a tangent is a line that touches a circle at exactly one point. It shares the name (from Latin “touching”) but is a different idea from the tangent ratio.
Cosine is adjacent over hypotenuse, and tan θ = sin θcos θ.
Quick check: 3 questions
Question 1 of 3
In a right triangle, the leg opposite θ is 8 and the leg adjacent to θ is 15. What is tan θ?
Show answer and explanation
Answer: C. Opposite over adjacent: 815. 158 flips it, and the fractions over 17 use the hypotenuse, which gives sine and cosine.
Question 2 of 3
What is tan 45°?
Show answer and explanation
Answer: B. In a 45-45-90 triangle the legs are equal, so opposite over adjacent is 1. √22 is sin 45°, 0 is tan 0°, and tangent is undefined at 90°.
Question 3 of 3
What is tan 90°?
Show answer and explanation
Answer: D.tan 90° = sin 90°cos 90° = 10, which is undefined. √3 is tan 60° and 1 is tan 45°.
Tangent: frequently asked questions
What is tangent in SOH CAH TOA?
TOA stands for Tangent = Opposite over Adjacent. In a right triangle, pick an acute angle; the opposite side is across from it and the adjacent side is the leg that touches it. The tangent is opposite divided by adjacent. SOH and CAH give sine (opposite over hypotenuse) and cosine (adjacent over hypotenuse).
How do you find an angle using tangent?
Divide the opposite leg by the adjacent leg, then use the inverse tangent, tan−1 (also called arctan), on a calculator in degree mode. For legs 5 (opposite) and 5 (adjacent), tan−1(1) = 45°. For 7 and 10, tan−1(0.7) ≈ 35°.
Why is tan 90 degrees undefined?
Because tan θ = sin θcos θ and cos 90° = 0, so it would mean dividing by zero. In a right triangle, as the angle approaches 90° the adjacent leg shrinks toward 0, and the tangent grows without bound. That is why the graph of y = tanx has a vertical asymptote at 90°.
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