Mathwords logoReference LibraryMathwords

Tangent

tangent
tan

The trig function tangent, written tan θ. tan θ equals Right triangle with angle θ, showing sin θ (opposite side) and cos θ (adjacent side), illustrating tan θ = sin θ / cos θ.. For acute angles, tan θ can be found by the SOHCAHTOA definition as shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = tan x is a periodic function with period π.

 

Two diagrams showing tan θ = opposite/adjacent (right triangle, SOHCAHTOA) and tan θ = y/x = slope of radius (unit circle).

 

 

See also

Inverse tangent, angle

Key Formula

tan⁡θ=oppositeadjacent=sin⁡θcos⁡θ\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}
Where:
  • θ\theta = The angle in a right triangle or on the unit circle
  • opposite\text{opposite} = The side of the right triangle across from the angle θ
  • adjacent\text{adjacent} = The side of the right triangle next to the angle θ (not the hypotenuse)

Worked Example

Problem: In a right triangle, the side opposite angle θ has length 3 and the side adjacent to angle θ has length 4. Find tan θ.
Step 1: Identify the opposite and adjacent sides relative to angle θ.
opposite=3,adjacent=4\text{opposite} = 3, \quad \text{adjacent} = 4
Step 2: Apply the tangent ratio from SOHCAHTOA: tangent equals opposite over adjacent.
tan⁡θ=oppositeadjacent=34\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
Step 3: Compute the decimal value.
tan⁡θ=0.75\tan\theta = 0.75
Answer: tan θ = 3/4 = 0.75

Another Example

This example uses the unit circle definition for a non-acute angle, showing how reference angles and quadrant signs determine the value of tangent beyond the first quadrant.

Problem: Find the exact value of tan 120°.
Step 1: Determine the reference angle. Since 120° is in the second quadrant, the reference angle is 180° − 120° = 60°.
reference angle=180°−120°=60°\text{reference angle} = 180° - 120° = 60°
Step 2: Recall that tan 60° = √3.
tan⁡60°=3\tan 60° = \sqrt{3}
Step 3: Determine the sign. In the second quadrant, sine is positive and cosine is negative, so tangent (sin/cos) is negative.
Quadrant II: tan⁡θ<0\text{Quadrant II: } \tan\theta < 0
Step 4: Combine the reference angle value with the correct sign.
tan⁡120°=−3\tan 120° = -\sqrt{3}
Answer: tan 120° = −√3

Frequently Asked Questions

What is the difference between tangent, sine, and cosine?
Sine gives the ratio of the opposite side to the hypotenuse, cosine gives the ratio of the adjacent side to the hypotenuse, and tangent gives the ratio of the opposite side to the adjacent side. Tangent can also be calculated as sine divided by cosine: tan θ = sin θ / cos θ.
Why is tangent undefined at 90° and 270°?
Tangent equals sin θ / cos θ. At 90° and 270°, cos θ = 0, which means you would be dividing by zero. Division by zero is undefined, so tan 90° and tan 270° do not exist. These angles correspond to vertical asymptotes on the graph of y = tan x.
What is the period of the tangent function?
The tangent function has a period of π (or 180°). This means tan θ = tan(θ + π) for every value of θ where tangent is defined. This is half the period of sine and cosine, which each have a period of 2π.

Tangent (tan) vs. Sine (sin)

Tangent (tan)Sine (sin)
DefinitionOpposite / AdjacentOpposite / Hypotenuse
Unit circle formulasin θ / cos θ (i.e., y/x)y-coordinate of the point on the unit circle
RangeAll real numbers (−∞, ∞)[−1, 1]
Periodπ (180°)2π (360°)
Undefined valuesAt 90° + 180°n (where cos θ = 0)Defined for all real numbers

Why It Matters

Tangent appears constantly in geometry, physics, and engineering whenever you need to relate vertical and horizontal distances. For example, if you know the angle of elevation to the top of a building and your distance from it, tangent lets you calculate the building's height. It is also essential in calculus, where the derivative of tan x = sec²x is a standard result, and in precalculus when analyzing periodic functions and their graphs.

Common Mistakes

Mistake: Confusing the opposite and adjacent sides when computing tan θ.
Correction: The opposite and adjacent sides depend on which angle you are looking at. The opposite side is always across from your chosen angle, and the adjacent side is the one touching your angle that is not the hypotenuse. Labeling the triangle clearly before computing avoids this error.
Mistake: Forgetting that tangent is undefined when cos θ = 0 (at 90°, 270°, etc.).
Correction: Since tan θ = sin θ / cos θ, you must check that cos θ ≠ 0 before evaluating. At θ = 90° + 180°n (for any integer n), tangent has vertical asymptotes and no defined value.

Related Terms