Right triangles, special ratios and trigonometry
Identify corresponding sides and use special triangles or sine, cosine and tangent to model a right triangle.
What you will tackle
- Apply the Pythagorean theorem and distance reasoning
- Use 45–45–90 and 30–60–90 side ratios
- Define sine, cosine and tangent through similar right triangles
- Model contextual right triangles and use complementary-angle relationships
Label relative to the selected angle
The hypotenuse lies opposite the right angle. For an acute angle θ, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. Similar right triangles with the same acute angle give the same ratios. Sine and cosine exchange roles for complementary angles.
Special triangles reduce calculation
A 45–45–90 triangle has sides s,s,s√2. A 30–60–90 triangle has sides s,s√3,2s opposite 30°,60°,90°. For a general right triangle use a² + b² = c² and check that c is the largest positive length. Coordinate distance follows the same theorem.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
A right triangle has legs 5 and 12. For the angle opposite the side of length 5, find the hypotenuse, sin θ, cos θ and tan θ.
Show a starting hint
The labels opposite and adjacent change with the chosen acute angle.
Compare with the explained answer
- The hypotenuse is √(25 + 144) = 13.
- Opposite is 5 and adjacent is 12.
- Use the three ratio definitions.
Answer / response criteria: 13; sin θ = 5/13, cos θ = 12/13, tan θ = 5/12.
Independent practice
Independent 1
Find the hypotenuse of a 45–45–90 triangle with leg 7, and both legs of a 30–60–90 triangle with hypotenuse 18. What is cos 60°?
Compare with the explained answer
- Equal legs 7 give hypotenuse 7√2.
- The short leg is 18/2 = 9 and the long leg is 9√3.
- At 60°, adjacent/hypotenuse is 9/18 = 1/2.
Answer / response criteria: 7√2; 9 and 9√3; cos 60° = 1/2.
Application practice
Application 1
A ladder 10 m long reaches a vertical wall at an angle of 60° to level ground. Find its height and distance from the wall. Explain why the ladder length is the hypotenuse.
Compare with the explained answer
- The wall and ground form the right angle, so the opposite side—the ladder—is the hypotenuse.
- Height = 10 sin 60° = 5√3 m.
- Ground distance = 10 cos 60° = 5 m.
Answer / response criteria: Height 5√3 m; distance 5 m.
What can you explain now?
A ladder 10 m long reaches a vertical wall at an angle of 60° to level ground. Find its height and distance from the wall. Explain why the ladder length is the hypotenuse.
Compare with the explanation
Height 5√3 m; distance 5 m.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide