New
New
Year 9

Tangent ratio

I can derive the tangent ratio from the sides of a right-angled triangle.

New
New
Year 9

Tangent ratio

I can derive the tangent ratio from the sides of a right-angled triangle.

Lesson details

Key learning points

  1. When the adjacent has a length of one, the opposite side has a length of tan⁡(θ).
  2. You can apply a scale factor to this triangle to find the scaled length of the opposite side.
  3. There is a multiplicative link between two similar right-angled triangles.
  4. There is a multiplicative link within each right-angled triangle.

Common misconception

tan(60°) is double tan(30°).

The values of tan of an angle do not scale linearly. From the unit circle, we see that an angle of 30° meets a tangent to the circle at a height of approx. 0.58 units, whilst an angle of 60° meets the same tangent at a height of approx. 1.73 units.

Keywords

  • 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.

  • Trigonometric ratios - The trigonometric ratios are ratios between each pair of lengths in a right-angled triangle.

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

Calculators can be used across all learning cycles to perform calculations using values found in the table of trig values or measured from the unit circle, however there is a bespoke learning cycle dedicated to introducing trigonometric values using a calculator.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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6 Questions

Q1.
In a right triangle, if the hypotenuse is 32.5 cm and a second side is 12.5 cm, what is the perimeter? (Use a calculator to help you.)
Correct answer: 75 cm
80 cm
70 cm
Q2.
What is the value of θ for sin(θ) = tan(θ) ?
Correct answer: 0ᵒ
45ᵒ
90ᵒ
Q3.
What is the value of θ for tan(θ) = 1
Correct answer: 45ᵒ
90ᵒ
30ᵒ
0ᵒ
Q4.
In a right triangle, if the hypotenuse is 32.5 cm and a second side is 12.5 cm, what is the area? (Use a calculator to help you.)
Correct answer: $$187.5 cm^2 $$
$$197.5 cm^2 $$
$$207.5 cm^2 $$
Q5.
What is the value of θ for sin(θ) = 1
Correct answer: 90ᵒ
45ᵒ
50ᵒ
30ᵒ
Q6.
What is the value of θ for sin(θ) = cos(θ) ?
Correct answer: 45ᵒ
90ᵒ
30ᵒ
0ᵒ

6 Questions

Q1.
Match the estimated values with the trigonometric functions below.
Correct Answer:sin(23°),0.39

0.39

Correct Answer:cos(39°),0.777

0.777

Correct Answer:tan(22°),0.404

0.404

Correct Answer:tan(45°),1

1

Q2.
For which value of θ is tan (θ) undefined?
Correct Answer: 90
Q3.
In a right triangle, if you are provided with θ and the opposite side, which function would you use to find the adjacent side?
Sine
Cosine
Correct answer: Tangent
Q4.
Which of these is a rearrangement of $$tan(θ)=\frac{opp}{adj}$$
$$tan(θ)=\frac{adj}{opp}$$
$$opp=\frac{tan(θ)}{adj}$$
Correct answer: $$adj=\frac{opp}{tan(θ)}$$
Q5.
Which of these is not an rearrangement of $$tan(θ)=\frac{opp}{adj}$$
$$adj=\frac{opp}{tan(θ)}$$
$$tan(θ)\times adj=opp$$
Correct answer: $$tan(θ)=\frac{adj}{opp}$$
Q6.
For a right triangle ABC, if angle CBA = 33ᵒ, and the adjacent side to that angle is 1 cm, what is the length of the side opposite angle CBA?
0.545 cm (3 d.p)
Correct answer: 0.649 cm (3 d.p)
1 cm