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      Checking and securing understanding of tangent ratio problems

      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 confuse the opposite and adjacent sides.

      The orientation of the right-angled triangle is not important. Identifying the known angle is what determines which is is the opposite and which is the adjacent.

      Teacher tip

      It may be helpful to introduce the tangent formula as a × tan(θ°) = opp, that is to say the length of the side opposite to an angle is equal to the length of the adjacent multiplied by the tangent of that angle, so that both rearrangements of the formula can be shown using a one-step division

      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.
      Match the names of the sides, relative to the angle $$\theta$$.

      An image in a quiz
      Correct Answer:a,opposite
      Correct Answer:b,adjacent
      Correct Answer:c,hypotenuse

      Q2.
      The two triangles are similar. What is the length of $$x$$?

      An image in a quiz
      Correct Answer: 1.46 cm, 1.46

      Q3.
      Triangle ABC and triangle DEF are similar. What is the length of $$x$$?

      An image in a quiz
      Correct Answer: 16 cm, 16, sixteen

      Q4.
      Work out the value of $$x$$ to 3 significant figures.

      An image in a quiz
      Correct Answer: 4.99, 4.99 cm

      Q5.
      Work out the value of $$x$$ to 1 decimal place.

      An image in a quiz
      Correct Answer: 8.6, 8.6 cm

      Q6.
      Work out the length of $$x$$, to 1 decimal place.

      An image in a quiz
      Correct Answer: 12.7 cm, 12.7

      6 Questions

      Q1.
      For a right-angled triangle, the tangent ratio is $$\tan(\theta)=\frac{\text{opp}}{\text{adj}}$$, where $$\theta$$ is the angle in degrees. Which of these are equivalent forms?

      $$\tan(\theta)=\frac{\text{adj}}{\text{opp}}$$
      Correct answer: $$\text{adj}\times\tan(\theta)=\text{opp}$$
      $$\text{adj}=\frac{\tan(\theta)}{\text{opp}}$$
      Correct answer: $$\theta=\arctan\left(\frac{\text{opp}}{\text{adj}}\right)$$
      Correct answer: $$\text{adj}=\frac{\text{opp}}{\tan(\theta)}$$

      Q2.
      Work out the length of $$x$$, to 1 decimal place.

      An image in a quiz
      Correct Answer: 11.7 cm, 11.7

      Q3.
      Work out the length of $$x$$, to 1 decimal place.

      An image in a quiz
      Correct Answer: 10.1 cm, 10.1

      Q4.
      Using the table of values, what is the value of $$\theta$$?

      An image in a quiz
      0
      5
      10
      Correct answer: 15
      20

      Q5.
      Match the variables to their values.

      An image in a quiz
      Correct Answer:opposite,33
      Correct Answer:adjacent,48
      Correct Answer:theta,34.50852299

      Q6.
      Work out the value of $$\theta$$, to the nearest degree.

      An image in a quiz
      Correct Answer: 42, forty two, 42 degrees

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