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.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide