Trig ratios of an acute angle – measuring height with elevation angle
MathematicsTrigonometryAges 14–15
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Trigonometry
Right triangle ABC, right-angled at A: drag vertex C to change the acute angle B = α, drag vertex A to change the adjacent side AB; sin, cos, tan and cot of α update instantly from the side ratios. In application mode, an observer stands a set distance from a building; drag the angle of elevation to find the building's height using tan α.
Lesson: Trigonometric ratios of an acute angle; using them to measure height
What it shows
In a right triangle ABC with the right angle at A, the ratios of the acute angle α at B are sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent and cot α = adjacent/opposite. Enlarging the triangle while keeping α fixed leaves every ratio unchanged, so the ratios depend only on the angle. The identities sin²α + cos²α = 1 and sin C = cos α for complementary angles are checked live. The application mode finds a building's height from the angle of elevation: h = d·tan α plus the observer's eye height.
How to use
In Right triangle mode, change the Acute angle α and watch the four ratios update; then keep α fixed and change the Adjacent side AB to show that the ratios do not change. Switch to Measure building height, set the distance, the angle of elevation and the Observer's eye height, and ask students to compute the height with tan α before reading the answer. Discuss what happens if the eye height is left out.
Parameters you can change
- Acute angle α 5–85 degrees
- Adjacent side AB (distance to the building) 1–20 m
- Mode Right triangle – trig ratios, Application: measuring a building's height
- Observer's eye height 1–2 m
Questions to explore
- Why is sin α always less than 1 for an acute angle?
- As α increases from 5° to 85°, how does tan α change?
- How large is the error in the building's height if you ignore the observer's eye height?
