Tangent Ratio (TOA)

Master the tangent ratio and learn how to find missing sides and angles using opposite and adjacent.

Advanced25 minLesson
Printable worksheet

Definition

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
The mnemonic TOA helps you remember:
  • T = Tangent
  • O = Opposite
  • A = Adjacent
For example, in a right triangle where the angle is 45°45°:
tan⁡(45°)=11=1\tan(45°) = \frac{1}{1} = 1
This means the opposite and adjacent sides are equal in length when the angle is 45°45°.

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What is the tangent ratio formula?

Worked Examples

You stand 5050 meters from a building. The angle of elevation to the top of the building is 32°32°. How tall is the building?

1

Identify the known values

Angle = 32°32°, Adjacent (distance from building) = 5050 m, Unknown = Opposite (height) → Set up the problem

2

Write the tangent formula

tan⁡(32°)=opposite50\tan(32°) = \frac{\text{opposite}}{50} → Formula ready

3

Find tan⁡(32°)\tan(32°)

tan⁡(32°)≈0.625\tan(32°) \approx 0.625

4

Solve for the opposite side

opposite=50×0.625=31.25\text{opposite} = 50 \times 0.625 = 31.25 m → 31.2531.25 meters

Common Mistakes

Confusing tangent with sine or cosine

Why it's wrong: Tangent uses opposite and adjacent, while sine and cosine involve the hypotenuse. Tangent does not use the hypotenuse at all.

Correct: Remember TOA: Tangent = Opposite / Adjacent. No hypotenuse needed!

Dividing adjacent by opposite instead of opposite by adjacent

Why it's wrong: The order matters! Tangent is always opposite divided by adjacent, not the reverse.

Correct: Think: 'Opposite over Adjacent' - the opposite is on top of the fraction.

Not recognizing that tan⁡(90°)\tan(90°) is undefined

Why it's wrong: At 90°90°, the adjacent side has length 00, and division by zero is undefined.

Correct: Remember: tan⁡(θ)\tan(\theta) approaches infinity as θ\theta approaches 90°90°. At exactly 90°90°, it's undefined.

Confusing angle of elevation with angle of depression

Why it's wrong: Both use tangent, but angle of elevation looks up while angle of depression looks down.

Correct: Angle of elevation: from horizontal up to the object. Angle of depression: from horizontal down to the object. Both angles equal when measured from the horizontal.

Interactive Visual

Right Triangle Trigonometry

θ =30°
5°45°85°
sin(θ)
0.5
Opp / Hyp
cos(θ)
√3/2
Adj / Hyp
tan(θ)
√3/3
Opp / Adj

Explore how the tangent ratio (opposite/adjacent) changes with the angle.

Interactive Sandbox

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Practice Problems

16 problems
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What is the tangent ratio formula?

Why It Matters

The tangent ratio is crucial for problems involving height and distance without needing the hypotenuse:
  • Surveying: Calculating the height of buildings, trees, or mountains from ground level
  • Navigation: Finding distances when you know the angle and one leg of a right triangle
  • Construction: Determining roof pitches, ramp slopes, and staircase angles
  • Aviation: Calculating descent angles and runway approaches
  • Photography: Understanding field of view and perspective angles
Whenever you know the opposite and adjacent sides (or need to find one from the other), tangent is your go-to ratio.

Real World Applications

Measuring Heights Without Climbing

Surveyors and engineers use the tangent ratio to measure the heights of tall structures without physically climbing them.

Example:

To find the height of a cell tower, a surveyor stands 3030 meters away and measures the angle of elevation as 65°65°. Using tan⁡(65°)≈2.14\tan(65°) \approx 2.14, the height is approximately 30×2.14=64.330 \times 2.14 = 64.3 meters.

1Try It Yourself

You want to find the height of a flagpole. Standing 2020 meters from its base, you measure the angle of elevation to the top as 35°35°.

How tall is the flagpole?

Step 1: Write the mathematical expression

Use tangent: height = 20×tan⁡(35°)20 \times \tan(35°)

Roof Pitch and Construction

Builders use tangent to calculate the pitch (slope) of a roof, which determines how steep it is.

Example:

A roof rises 44 feet for every 1212 feet of horizontal run. The pitch angle is tan⁡−1(4/12)=tan⁡−1(0.333)≈18.4°\tan^{-1}(4/12) = \tan^{-1}(0.333) \approx 18.4°.

2Try It Yourself

A roof has a rise of 66 feet over a run of 88 feet. What is the angle of the roof pitch?

What is the pitch angle?

Step 1: Write the mathematical expression

Calculate: tan⁡−1(6/8)\tan^{-1}(6/8)

Key Takeaways

  • 1Tangent equals opposite divided by adjacent: tan⁡(θ)=oppadj\tan(\theta) = \frac{\text{opp}}{\text{adj}}
  • 2Remember TOA: Tangent = Opposite / Adjacent
  • 3To find the opposite: multiply adjacent by tan⁡(θ)\tan(\theta)
  • 4To find the adjacent: divide opposite by tan⁡(θ)\tan(\theta)
  • 5To find the angle: use inverse tangent θ=tan⁡−1(oppadj)\theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)
  • 6Special values: tan⁡(0°)=0\tan(0°) = 0, tan⁡(30°)=13\tan(30°) = \frac{1}{\sqrt{3}}, tan⁡(45°)=1\tan(45°) = 1, tan⁡(60°)=3\tan(60°) = \sqrt{3}, tan⁡(90°)\tan(90°) is undefined

Frequently Asked Questions

At 45°45°, the right triangle is isoceles (excluding the right angle), so the opposite and adjacent sides are equal. Equal divided by equal is 11.
At 45°45°, the right triangle is isoceles (excluding the right angle), so the opposite and adjacent sides are equal. Equal divided by equal is 11.
At 90°90°, the adjacent side shrinks to zero length. Since tangent equals opposite/adjacent, we would be dividing by zero, which is undefined. As the angle approaches 90°90°, tangent grows toward infinity.
Tangent is the ratio of sine to cosine: tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}. This is because opp/hypadj/hyp=oppadj\frac{\text{opp/hyp}}{\text{adj/hyp}} = \frac{\text{opp}}{\text{adj}}.
tan⁡−1\tan^{-1} is the inverse tangent function, also called arctangent (arctan). It answers: 'What angle has this tangent value?' For example, tan⁡−1(1)=45°\tan^{-1}(1) = 45° because tan⁡(45°)=1\tan(45°) = 1.

Glossary

Tangent
The ratio of the opposite side to the adjacent side in a right triangle: tan⁡(θ)=oppadj\tan(\theta) = \frac{\text{opp}}{\text{adj}}
Opposite side
The side of a right triangle that is across from (opposite to) the reference angle
Adjacent side
The side of a right triangle that is next to the reference angle (not the hypotenuse)
Inverse tangent
The function tan⁡−1\tan^{-1} (arctan) that finds an angle when you know its tangent value
Angle of elevation
The angle measured upward from the horizontal to a line of sight to an object above
Angle of depression
The angle measured downward from the horizontal to a line of sight to an object below

Formula Card

Tangent definition

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

The ratio of the opposite side to the adjacent side

Find opposite

opposite=adjacent×tan⁡(θ)\text{opposite} = \text{adjacent} \times \tan(\theta)

Multiply adjacent by tangent to find the opposite side

Find adjacent

adjacent=oppositetan⁡(θ)\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}

Divide opposite by tangent to find the adjacent side

Find angle

θ=tan⁡−1(oppositeadjacent)\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)

Use inverse tangent (arctan) to find the angle

Special value: tan(0)

tan⁡(0°)=0\tan(0°) = 0

Tangent of 0 degrees equals 0

Special value: tan(30)

tan⁡(30°)=13≈0.577\tan(30°) = \frac{1}{\sqrt{3}} \approx 0.577

Tangent of 30 degrees equals 1 over square root of 3

Special value: tan(45)

tan⁡(45°)=1\tan(45°) = 1

Tangent of 45 degrees equals exactly 1

Special value: tan(60)

tan⁡(60°)=3≈1.732\tan(60°) = \sqrt{3} \approx 1.732

Tangent of 60 degrees equals square root of 3

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