Lesson 4 — The Tangent Ratio
Big idea: The tangent ratio compares the two legs of a right triangle, so it can turn one angle and one leg into the other leg.
Words to know
| Word | Meaning |
|---|---|
| Tangent ratio | For an acute angle of a right triangle, the ratio of the length of the opposite leg to the length of the adjacent leg. |
| Opposite leg | The leg across from the angle being used. |
| Adjacent leg | The leg next to the angle being used (not the hypotenuse). |
| Angle of elevation | The angle measured upward from a horizontal line to a line of sight. |
| Angle of depression | The angle measured downward from a horizontal line to a line of sight. |
| Inverse tangent | The operation that finds an angle when its tangent is known. |
Concepts to know
- Definition: tan A = (opposite leg) ÷ (adjacent leg).
- For a fixed acute angle, the tangent is the same in every right triangle with that angle, because the triangles are similar.
- The tangent uses only the two legs; it never uses the hypotenuse.
- To find an angle from a tangent, use the inverse tangent key.
- In word problems, the angle of elevation or depression is usually one of the acute angles; draw the horizontal line first.
- The angle of depression from the top of a tall object equals the angle of elevation from the point below, because the two horizontal lines are parallel.
Formulas / rules
- tan A = opposite ÷ adjacent
- opposite = adjacent · tan A
- adjacent = opposite ÷ tan A
- A = tan⁻¹(opposite ÷ adjacent)
- tan 45° = 1
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a leg from an angle and the other leg | Multiply or divide using tan A, depending on whether the unknown is opposite or adjacent. |
| Find an angle from two legs | Divide opposite by adjacent, then use inverse tangent. |
| Solve a height or distance problem | Sketch the right triangle; label the angle and the two legs; write tan A = opp/adj. |
| Use an angle of elevation | Put the angle at the ground between the horizontal and the line of sight. |
| Use an angle of depression | Put the angle at the top between the horizontal and the line of sight. |
| Check an answer | The leg opposite the larger acute angle should be longer. |
Watch out
- Opposite and adjacent depend on which angle you are using — relabel if you switch angles.
- Never put the hypotenuse into the tangent ratio.
- Set your calculator to degree mode.
- Round only at the end, not at every step.
- Do not confuse the angle of elevation with the angle inside the triangle at the top of a building.