Choose a ratio from the sides, not habit
After drawing the right triangle, name the known side type and required side type relative to the observation angle. Their pair determines sin, cos or tan.
Learning goals
- Identify known, required and relevant right-triangle sides.
- Select sin, cos or tan from the known and required sides.
- Form and solve a one-triangle height or distance equation.
- Check units, positivity and diagram reasonableness.
Concept lesson
Diagram first
Mark vertical height, horizontal distance, sight line, right angle and observation angle before assigning O, A and H.
Exact standard values
tan30=1/sqrt(3), tan45=1 and tan60=sqrt(3); use exact forms unless an approximation is explicitly supplied.
Known-required side selector
Use tan for height and horizontal distance, sin for height and line of sight, and cos for horizontal distance and line of sight. This selection avoids calculating an unused third side.
Vertical difference versus total height
The triangle's opposite side is the vertical difference between observer eye level and target. Add observer height to obtain total target height, or subtract a known platform height when the unknown is only an upper segment.
Find height from ground distance
At horizontal distance 30 m and elevation 30 degrees, tan30=h/30. Hence h=30/sqrt(3)=10sqrt(3) m. The height and ground distance make tan the direct ratio.
Find distance from known height
For windmill height 40sqrt(3) m viewed at 30 degrees, tan30=40sqrt(3)/d, so d=120 m. Check that a small elevation angle produces a ground distance greater than the height.
Include eye height
If an observer's eyes are e metres above ground and the triangle gives vertical difference v, total object height is v+e. Label the eye-level horizontal so e is not hidden inside the triangle.
Select without solving
For six diagrams, name known and required side types and choose the direct ratio. Explain why the other two ratios introduce missing quantities.
Reasonableness and units audit
Audit proposed answers for positive length, common units, hypotenuse-longest condition and expected size at 30,45,60 degrees. Correct the earliest invalid step.
Ladder, shadow and slide models
Decide whether the sloping object is a line of sight or physical hypotenuse, select the direct ratio and solve a ladder, shadow or playground-slide length problem with units.
Two-mark solution structure
Draw or reference the triangle, state the chosen ratio, substitute the standard value, solve, include any height offset and report a positive answer with units.
Direct heights-and-distances mastery
Solve a height, ground-distance, ladder and eye-height problem, choosing direct ratios and auditing exact values, units and reasonableness.