KBAT Problems
KBAT Problem: Finding a Height You Cannot Measure
A KBAT problem using angles of elevation to find a height indirectly, the classic real use of trigonometry.
Draw the right triangle
To find the height of a tower from a distance and an angle of elevation, sketch the situation as a right-angled triangle with the height opposite the angle. Choosing tangent, opposite over adjacent, is the KBAT decision.
Check the calculator mode
Because trig in SPM works in degrees, confirm the calculator shows DEG before computing. A common lost mark here is a radian-mode answer that is numerically wrong but looks reasonable.
Understand the problem
From a point on level ground 40 m from the base of a flagpole, the angle of elevation to the top of the pole is 35°. Find the height of the pole.
What it really asks: model the situation as a right-angled triangle, notice you know the adjacent side and the angle and want the opposite side, and choose the tangent ratio to link them.
Plan and solve
- Sketch a right-angled triangle: the 40 m ground distance is the adjacent side, the pole height h is the opposite side, and the angle is 35°.
- The ratio linking opposite and adjacent is tangent, so tan 35° = h / 40.
- Rearrange: h = 40 × tan 35°.
- With the calculator in degree mode, tan 35° ≈ 0.7002, so h ≈ 40 × 0.7002 ≈ 28.0 m.
Check and a variant
Check the answer is sensible: the angle is under 45°, so the height should be less than the 40 m distance, and 28.0 m is, which is a good quick sanity check. A variant the examiner could add: the observer's eye is 1.5 m above the ground rather than at ground level.
Then the pole's full height is the 28.0 m above eye level plus the 1.5 m eye height, giving 29.5 m, a small adjustment that catches careless answers.
Frequently asked questions
How do I choose between sine, cosine and tangent?
Label the sides relative to the angle as opposite, adjacent and hypotenuse, then see which two you have or want. Opposite with adjacent means tangent, opposite with hypotenuse means sine, adjacent with hypotenuse means cosine.
Here you have the adjacent and want the opposite, so tangent is the correct choice.
Why does my calculator sometimes give a strange trig answer?
Almost always because it is in radian or gradian mode instead of degrees. SPM trigonometry works in degrees, so confirm the screen shows DEG before you compute.
A radian-mode answer looks like a normal number but is numerically wrong, and it is a frequent and avoidable lost mark in this topic.
When does the observer's eye height matter?
It matters whenever the angle is measured from eye level rather than the ground, which the wording will tell you. In that case the tangent calculation gives only the height above the eye, so you add the eye height to get the full height.
Read the question carefully, because this small step separates a full-mark answer from a near miss.