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      Problem solving with right-angled trigonometry

      Lesson details

      Learning outcome

      I can use my knowledge of right-angled trigonometry to solve problems.

      Key learning points

      1. Sometimes an answer may be best left in an exact form
      2. When dealing with right-angled trigonometry, it is important to look at the information you have and can deduce
      3. Consider whether Pythagoras' theorem or trigonometric ratios are more efficient to use

      Keywords

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

      • Sine function - The sine of an angle (sin(θ°)) is the y-coordinate of point P on the triangle formed inside the unit circle.

      • Cosine function - The cosine of an angle (cos(θ°)) is the x-coordinate of point P on the triangle formed inside the unit circle.

      • 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.

      Common misconception

      Pupils may not be confident in knowing whether to apply Pythagoras' theorem or a trigonometric ratio.

      Encourage pupils to label the diagram with all the information they have and then consider what they can deduce.

      Teacher tip

      You may wish to provide square grids for pupils to explore the area question in the first learning cycle.

      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.
      Work out the length of the hypotenuse, to 1 decimal place.

      An image in a quiz
      Correct Answer: 56.3 cm, 56.3

      Q2.
      Work out the length of the missing side of this right-angled triangle, to 1 decimal place.

      An image in a quiz
      Correct Answer: 8.5 cm, 8.5

      Q3.
      Work out the length of the line segment AB, where A(2, 6) and B(5, 10).

      Correct Answer: 5, five, 5 units

      Q4.
      Work out $$x$$ to 1 decimal place using the tangent ratio.

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      Correct Answer: 8.7, 8.7 cm

      Q5.
      Work out the size of the angle, $$x$$.

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      Correct Answer: 30, 30 degrees

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

      An image in a quiz
      Correct Answer: 3.6 cm, 3.6

      6 Questions

      Q1.
      ABCD is a square on a grid, where each square is 1 unit. Work out the length of BC, to 1 decimal place.

      An image in a quiz
      Correct Answer: 5.4, 5.4 units

      Q2.
      Which of these calculations are correct for finding the area of the square ABCD?

      An image in a quiz
      Correct answer: $$7^2-4\times\left(\frac{2\times5}{2}\right)$$
      $$2\times10+3^2$$
      Correct answer: $$2^2+5^2$$
      Correct answer: $$(\sqrt{2^2+5^2})^2$$
      $$4\times\left(\frac{2\times5}{2}\right)-7^2$$

      Q3.
      Match the edge with the correct calculation.

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      Correct Answer:AE,$$=\frac{16}{\tan(71.6^\circ)}$$
      Correct Answer:DC,$$=\frac{16}{\sin(71.6^\circ)}$$
      Correct Answer:FG,$$=HC+\frac{16}{\tan(71.6^\circ)}$$

      Q4.
      Given that this is an isosceles triangle, calculate the angle marked $$x$$ to 1 decimal place.

      An image in a quiz
      Correct Answer: 74.7, 74.7 degrees

      Q5.
      The area of this isosceles triangle is $$\text{cm}^2$$ to 3 significant figures.

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      Correct Answer: 118

      Q6.
      Given that AB is the diameter of the circle, the area of the circle is $$\text{cm}^2$$ to 3 significant figures.

      An image in a quiz
      Correct Answer: 331

      To help you plan your 10 maths lesson on: Problem solving with right-angled trigonometry, download all teaching resources for free and adapt to suit your pupils' needs...