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Using the tangent ratio

Lesson details

Learning outcome

I can use the tangent ratio to find the missing side or angle in a right-angled triangle.

Key learning points

  1. The tangent ratio involves the opposite, adjacent and the angle.
  2. If you know the length of the opposite and the size of the angle, you can use the tangent ratio.
  3. If you know the length of the adjacent and the size of the angle, you can use the tangent ratio.
  4. If you know the length of the opposite and adjacent, you can use the tangent ratio.

Keywords

  • Trigonometric functions - Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle containing the angle.

  • Tangent function - The tangent of an angle (tan(θ°)) is the y-coordinate of point Q of the triangle which extends from the unit circle.

  • Adjacent - The adjacent side of a right-angled triangle is the side which is next to both the right angle and the marked angle.

  • Opposite - The opposite side of a right-angled triangle is the side which is opposite the marked angle.

Common misconception

Pupils may get the fraction (opposite/adjacent) inverted and this will still return an answer so they may believe they are correct.

Remind pupils to sense check their answers. The larger the angle, the longer the opposite side will be.

Teacher tip

Ask pupils to sketch, perhaps on mini-whiteboards, a right-angled triangle which satisfies tan(𝜃°) = 0.4 or tan(𝜃°) = 2, to embed that the 0.4 or 2 is the proportions of the opposite and the adjacent for some given angle 𝜃.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
except where otherwise stated. See Oak's terms & conditions
(Collection 2).

Lesson video

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Prior knowledge starter quiz

6 Questions

Q1.
In a right triangle, if the hypotenuse is 25 cm and a second side is 20 cm, what is the perimeter? (Use a calculator to help you.)

Correct answer: 60 cm
66 cm
70 cm

Q2.
In a right triangle, if the hypotenuse is 125 cm and a second side is 100 cm, what is the area? (Use a calculator to help you.)

$$3700 cm^2 $$
Correct answer: $$3750 cm^2 $$
$$3520 cm^2 $$

Q3.
What is the value of cos(37ᵒ) to 2 d.p. ?

Correct answer: 0.80
0.70
0.90

Q4.
Which of these is a correct formula for using the Cosine function?

Correct answer: $$cos(x) = \frac{adj}{hyp}$$
$$cos(x) = \frac{opp}{hyp}$$
$$cos(x) = \frac{hyp}{adj}$$
$$cos(x) = \frac{adj}{opp}$$

Q5.
Which of these is a correct formula for using the Sine function?

Correct answer: $$sin(x) = \frac{opp}{hyp}$$
$$sin(x) = \frac{opp}{adj}$$
$$sin(x) = \frac{adj}{hyp}$$
$$sin(x) = \frac{hyp}{opp}$$

Q6.
Which of these is a correct formula for using the Tangent function?

Correct answer: $$tan(x) = \frac{opp}{adj}$$
$$tan(x) = \frac{adj}{opp}$$
$$tan(x) = \frac{opp}{hyp}$$
$$tan(x) = \frac{hyp}{adj}$$

6 Questions

Q1.
Using triangle ABC, if θ = 38ᵒ and BC = 14 cm, what is the length of the adjacent side AC to 1 d.p. ?

An image in a quiz
Correct Answer: 17.9, 17.9cm

Q2.
What value of $$y$$ will produce the same result for tan($$y$$) and cos(90ᵒ)?

Correct answer: 0ᵒ
45ᵒ
90ᵒ
60ᵒ

Q3.
Using triangle ABC, if θ = 18ᵒ and BC = 10 cm, what is the length of the adjacent side AC to 1 d.p. ?

An image in a quiz
Correct Answer: 30.8, 30.8cm

Q4.
What value of $$y$$ will produce the same result for tan($$y$$) and cos(0ᵒ)?

0ᵒ
Correct answer: 45ᵒ
90ᵒ
60ᵒ

Q5.
Using triangle ABC, if θ =26ᵒ and AC = 10 cm, what is the length of BC to 1 d.p. ?

An image in a quiz
Correct Answer: 4.9, 4.9cm

Q6.
Using triangle ABC, if θ =35ᵒ and AC = 10 cm, what is the length of BC to 1 d.p. ?

An image in a quiz
Correct Answer: 7.0, 7.0cm

To help you plan your 9 maths lesson on: Using the tangent ratio, download all teaching resources for free and adapt to suit your pupils' needs...