KS3 · Maths

Right-angled triangle trigonometry: tangent

Squash a right-angled triangle until one side is exactly 1 unit long. The side opposite the angle is now a number with a name: tan θ.

Maths · Trigonometry

Drag the top corner
30°1.000.58OA1tan 60° ≈ 1.73tan 75° = 3.732T

Angle θ 30°. Adjacent OA 1.00. Opposite AT = tan θ 0.58. Classification: Right-angled at A. Relationship: Adjacent = 1 unit, so the opposite side AT has length tan θ.

Angle θ30°Adjacent OA1.00Opposite AT = tan θ0.58drag T ↑↓ and read the opposite

Right-angled at A

Adjacent = 1 unit, so the opposite side AT has length tan θ.

O is the corner with θ. A is the right-angle corner. The adjacent side OA is fixed at 1 unit. Drag T up and down the vertical line and watch the opposite side AT.

Watch out: The angle doubles from 30° to 60°, but the opposite goes from about 0.58 to about 1.73. That is not double 0.58.

Maths · Going the other way

Find the scale factor from the opposite
0.01.63.24.86.48.09.611.212.814.416.0drag until the opposite hits 16 →

How much of the triangle: 4/10. Opposite (cm) 6.4. Scale factor (× the adjacent-1 triangle) ×9.14. Adjacent (cm) 9.14. Opposite ÷ adjacent 0.700

Opposite (cm)6.4Scale factor (× the adjacent-1 triangle)×9.14Adjacent (cm)9.14Opposite ÷ adjacent0.700

The opposite is the side you know: 16 cm, with θ = 35°. Drag along the bar to grow the adjacent-1 triangle until its opposite reaches 16, then read the adjacent. One chip never changes.

Maths · Algebra

One equation, three forms

Start from adjacent × tan θ = opposite. Pick the form whose subject is the thing you want to find.

GoalWrite the link so the opposite side is on its own.
1
adjacent × tan θ = opposite

Inside any right-angled triangle, the adjacent multiplied by tan θ gives the opposite (θ below 90°).

2

Step 1 of 2

Inside any right-angled triangle, the adjacent multiplied by tan θ gives the opposite (θ below 90°).

Watch out: To find the adjacent you divide by tan θ. Multiplying by it finds the opposite instead.

Your turn

Supply the missing steps

A right-angled triangle has θ = 61° and an adjacent side of 100 cm. Find the opposite side.

  1. Label first: the adjacent is 100 cm, θ = 61°, and the opposite is the side we want.
  2. missing step
Which line is step 2?

Finding an angle

Which angle has tan θ = 0.714?

A triangle has tan θ = 0.714. Picture the unit circle: go up the tangent to a height of 0.714, then draw back to the centre. Estimate the angle θ.

Your estimate

45°

0°90°

Check the working

Where does this answer go wrong?

In a right-angled triangle the side opposite θ is 2.184 cm and the side adjacent to θ is 6 cm. Find θ.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Pin one side at 1, and the other side turns into a number you can look up.

What you need to know

  • The opposite side is across from the marked angle θ. The adjacent side is next to both θ and the right angle.
  • Fix the adjacent at one unit. Then the opposite side's length is tan θ, short for the tangent of θ.
  • Have a goPip squeezes a triangle until its adjacent side is exactly 1 unit. The opposite side now measures 0.9 units. What is tan θ?

    tan θ = 0.9

    With the adjacent at one unit, the opposite side's length is tan θ, so you just read it off.

  • Picture the unit circle: the opposite side sits on a tangent to it, and tan θ is the height you climb.
  • With the adjacent at 1, a bigger angle means a longer opposite, and tan θ can pass 1: tan 75° = 3.732.
  • Have a goZed's calculator says tan 80° is smaller than tan 20°. Zed swears the calculator never lies. What do you tell Zed?

    Something is wrong: tan 80° should be the bigger one.

    With the adjacent at 1, a bigger angle means a longer opposite, and tan θ can pass 1: tan 75° = 3.732.

  • It isn't in step with the angle: tan 30° ≈ 0.58 but tan 60° ≈ 1.73, not double.
  • A right-angled triangle with angle θ is a scaled copy of the adjacent-1 one, so opposite = adjacent × tan θ.
  • Have a goA right-angled triangle has an adjacent side of 5 cm and a tan θ of 1.2. How long is the opposite side?

    6 cm

    opposite = adjacent × tan θ, so 5 × 1.2 = 6. It's the adjacent-1 triangle scaled up by 5.

  • Rearrange for what you want: adjacent = opposite ÷ tan θ, and tan θ = opposite ÷ adjacent.
  • To find an angle, work out opposite ÷ adjacent, then press tan⁻¹ (usually Shift + tan) or use the table.
  • Check your calculator shows D (degrees), and round only at the end of the calculation.
  • Sense-check: for the same adjacent, a larger angle gives a longer opposite. A flipped fraction still gives an answer.

The big picture

Fix the adjacent side at one unit and the opposite side's length is tan θ. Any right-angled triangle is a scaled copy of that one, so opposite = adjacent × tan θ, adjacent = opposite ÷ tan θ and tan θ = opposite ÷ adjacent. Use tan⁻¹ to find an angle, keep the calculator in degrees, round only at the end and sense-check.

Key points

1Opposite and adjacent are named from the marked angle θ. The adjacent also touches the right angle.
2With the adjacent at one unit, the opposite side is tan θ. Every right-angled triangle with that angle is a scaled copy.
3The link inside a triangle is opposite = adjacent × tan θ. Rearranged: adjacent = opposite ÷ tan θ and tan θ = opposite ÷ adjacent.
4tan θ does not grow in step with θ, and it can be greater than 1.
5Find an angle with tan⁻¹ of (opposite ÷ adjacent). A unit-circle drawing is only an estimate.
6Degrees mode, rounding at the end and the sense-check on the adjacent keep answers trustworthy.

Worked example

Problem

A right-angled triangle has an adjacent side of 10.2 cm and θ = 50°. Use a ratio table and the table value tan 50° = 1.192 to find the opposite side.

⚠ Watch out

Flipping the fraction: writing tan θ = adjacent ÷ opposite. The calculator still gives an answer, so nothing warns you. Sense-check instead: for the same adjacent, a larger angle must give a longer opposite.

🧠

Memory hook

Stand 1 unit from a very tall vertical pole and look up at angle θ. The height where your line of sight hits the pole is tan θ.

✓

Check yourself

A right-angled triangle has an adjacent side of 4 cm and θ = 65°. Use the table value tan 65° = 2.145 to find the opposite side. (Answer: 4 × 2.145 = 8.58 cm.)

Flashcards

(14)
In a right-angled triangle, which side is the opposite?
The side across from the marked angle θ.
Which side is the adjacent?
The side next to both the right angle and the marked angle θ.
What is tan θ when the adjacent side is one unit long?
The length of the opposite side. It lies on a tangent to the unit circle.
Why are two right-angled triangles with the same angle linked by a scale factor?
They are similar, so every side is multiplied by the same scale factor.
What is the tangent ratio?
tan θ = opposite ÷ adjacent
Finding the opposite: multiply or divide?
Multiply the adjacent by tan θ.
Finding the adjacent: multiply or divide?
Divide the opposite by tan θ.
Which calculator button turns a ratio back into an angle?
tan⁻¹ (arctan), usually reached with the Shift key: θ = tan⁻¹(opposite ÷ adjacent).
Does doubling an angle double its tangent?
No. tan 30° ≈ 0.58 but tan 60° ≈ 1.73. The values do not scale linearly.
Can tan θ be greater than 1?
Yes. For example tan 75° = 3.732.
What should the calculator show before you use the tan button?
D, which means degrees mode.
When do you round in a tangent calculation?
Only at the end, to a suitable number of decimal places or significant figures.
What sense-check can catch a flipped fraction?
For the same adjacent, a larger angle gives a longer opposite. If your answers disagree with that, check which way up the fraction is.
How reliable is an angle read from the unit circle?
It is only an estimate. A table or tan⁻¹ on a calculator is more accurate.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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