12.6 Trigonometry: the Tangent Ratio

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: the tangent ratio, finding sides and angles, and angles of elevation and depression. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Know that tan θ = opposite ÷ adjacent
• Use tan θ to find the opposite or the adjacent side
• Use tan⁻¹ to find an angle, and know that tan 45° = 1
• Solve problems with angles of elevation and depression

Key words

• Tangent (tan): For a marked angle in a right-angled triangle, tan θ = opposite ÷ adjacent.
• TOA: Memory aid: Tangent = Opposite over Adjacent.
• Angle of elevation: The angle measured up from the horizontal.
• Angle of depression: The angle measured down from the horizontal.

The tangent ratio

Tangent uses the two shorter sides: the opposite and the adjacent. It does not use the hypotenuse at all.

Key fact: tan θ = O ÷ A so O = A × tan θ, A = O ÷ tan θ, θ = tan⁻¹(O ÷ A)

Worked example: tan 45°

A right-angled triangle has two equal shorter sides. What is the angle θ?
O = A, so tan θ = O ÷ A = 1. The triangle is isosceles, so both angles are 45°.

Answer: θ = 45° and tan 45° = 1

Finding a side

If the unknown is the opposite side, multiply. If the unknown is the adjacent side, it is on the bottom, so divide.

Worked example: the adjacent side

The side opposite a 57° angle is 8 cm. Find the adjacent side.
tan 57° = 8 ÷ A, so A = 8 ÷ tan 57° = 5.195… = 5.20 cm

Answer: 5.20 cm

Finding an angle

Work out O ÷ A, then use tan⁻¹ (SHIFT, tan).

Worked example: tan⁻¹

The opposite side is 5 cm and the adjacent side is 12 cm. Find θ.
tan θ = 5 ÷ 12 = 0.4166… θ = tan⁻¹(5 ÷ 12) = 22.619…° = 22.6°

Answer: 22.6°

• sin θ = O ÷ H; Sides used opposite and hypotenuse; Memory aid SOH
• cos θ = A ÷ H; Sides used adjacent and hypotenuse; Memory aid CAH
• tan θ = O ÷ A; Sides used opposite and adjacent; Memory aid TOA

Angles of elevation and depression

Both are measured from the horizontal. They are equal, because they are alternate angles between parallel horizontal lines. Heights and ground distances are usually the opposite and adjacent sides, so tan is often the ratio to use.

Worked example: elevation

Priya stands 30 m from the foot of a tower on level ground. The angle of elevation of the top is 41°. How tall is the tower?
height = 30 × tan 41° = 26.079… = 26.1 m

Answer: 26.1 m

Worked example: depression

From the top of a 55 m cliff, the angle of depression of a buoy is 23°. How far is the buoy from the foot of the cliff?
The angle at the buoy is also 23°. tan 23° = 55 ÷ d, so d = 55 ÷ tan 23° = 129.57… = 130 m (3 s.f.)

Answer: 130 m

Common mistakes

• Using tan when the hypotenuse is involved. tan uses only O and A.
• Multiplying when the adjacent side is the unknown. Then A = O ÷ tan θ.
• Measuring an angle of depression from the vertical instead of the horizontal.
• Forgetting to add the height of a person’s eyes when the angle is measured from eye level.

Revision questions

Try these, then check your answers below each one.

Question 1. Use your calculator to work out tan 55°, correct to 4 decimal places.

Answer: 1.4281

Working: tan 55° = 1.42814… ≈ 1.4281.

Question 2. The side adjacent to a 40° angle is 8 cm. Work out the side opposite the 40° angle, correct to 3 s.f.

Answer: 6.71 cm

Working: 8 tan 40° = 6.7128… ≈ 6.71 cm.

Question 3. The side opposite a 62° angle is 10 cm. Work out the side adjacent to the 62° angle, correct to 3 s.f.

Answer: 5.32 cm

Working: 10 ÷ tan 62° = 5.3170… ≈ 5.32 cm.

Related lessons

• Previous lesson: 12.5 Trigonometry: the Cosine Ratio
• Next lesson: 12.7 Finding Lengths and Angles Using Trigonometry
• In this chapter: 12.1 Pythagoras' Theorem 1
• In this chapter: 12.2 Pythagoras' Theorem 2
• In this chapter: 12.3 Trigonometry: the Sine Ratio 1

Continue in the full course

This lesson is section 12.6 (EGCMF1206) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.