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      Using the sine ratio

      Lesson details

      Learning outcome

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

      Key learning points

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

      Keywords

      • Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.

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

      • Trigonometric function - 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.

      Common misconception

      The sine formula is only used to find the length of a side opposite an angle.

      Whilst the sine formula can be used to find the length of a side opposite an angle, a rearrangement of the formula also allows us to find the length of the hypotenuse given the opposite side. The arcsine function allows us to find the angle, itself.

      Teacher tip

      It may be helpful to introduce the sine formula as h × sin(θ°) = opp, that is to say the length of the side opposite an angle is equal to the length to the hypotenuse multiplied by the sine 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.
      In a right triangle, if the hypotenuse is 15 cm and a second side is 9 cm, what is the perimeter? (Use a calculator to help you.)

      Correct answer: 36 cm
      32 cm
      41 cm

      Q2.
      What is the value of θ for cos(θ) = 0

      Correct answer: 90ᵒ
      45ᵒ
      30ᵒ
      0ᵒ

      Q3.
      In a right triangle, if the hypotenuse is 25 cm and a second side is 15 cm, what is the area? (Use a calculator to help you.)

      Correct answer: $$150 cm^2 $$
      $$140 cm^2 $$
      $$300 cm^2 $$

      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.
      In a right triangle, if you are provided with θ and the hypotenuse, which function would you use to find the opposite side?

      Correct answer: Sine
      Cosine
      Tangent

      Q6.
      In a right triangle, if θ = 35ᵒ and the opposite side is 7 cm, what is the length of the hypotenuse to the nearest whole number?

      Correct answer: 12 cm
      9 cm
      10cm

      6 Questions

      Q1.
      Using triangle ABC, if θ = 35ᵒ and BC = 5 cm, what is the length of the hypotenuse AB to 1 d.p. ?

      An image in a quiz
      Correct Answer: 8.7, 8.7 cm

      Q2.
      What value of $$x$$ will produce the same result for cos($$x$$) and sin(30ᵒ)?

      30ᵒ
      Correct answer: 60ᵒ
      90ᵒ

      Q3.
      Using triangle ABC, if θ = 40ᵒ and BC = 4.8 cm, what is the length of the hypotenuse AB to 1 d.p. ?

      An image in a quiz
      Correct Answer: 7.5, 7.5cm

      Q4.
      What value of $$y$$ will produce the same result for sin($$y$$) and cos(26ᵒ)?

      36ᵒ
      46ᵒ
      Correct answer: 64ᵒ
      54ᵒ

      Q5.
      Using triangle ABC, if θ = 50ᵒ and AB = 12 cm, what is the length of side BC to 1 d.p. ?

      An image in a quiz
      Correct Answer: 9.2, 9.2cm

      Q6.
      Using triangle ABC, if θ = 45ᵒ and AB = 18 cm, what is the length of side BC to 1 d.p. ?

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

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