ریاضی
Angle of Elevation and Angle of Depression
Both angles are measured from the horizontal, never the vertical, and the depression angle from a tower top equals the elevation angle from the ground below.
Angle of elevation and angle of depression
The angle of elevation is the angle from the horizontal up to your line of sight; the angle of depression is the angle from the horizontal down to it.
- دیگر نام
- Angles of elevation and depression, Elevation and depression angles
- طلبہ کہاں پڑھتے ہیں
- Grade 9 or Grade 10 right-angled trigonometry, and again in the height-and-distance questions on Edexcel 1MA1 Higher and Cambridge IGCSE 0580.
مختصر جواب
An angle of elevation is measured from the horizontal upwards to the line of sight; an angle of depression is measured from the horizontal downwards. Because the two horizontal lines in the diagram are parallel, the depression angle from the top of a tower equals the elevation angle from the point below.
ایک مثال
height = 40 × tan 32° = 25.0 m (3 s.f.)
You stand 40 m from the foot of a tower and the angle of elevation to the top is 32°. The horizontal 40 m is the adjacent side, the height is the opposite side, so tangent is the ratio you need: tan 32° = height ÷ 40. Multiplying gives 24.99, or 25.0 m to three significant figures. If the 32° was measured from eye level 1.6 m above the ground, the tower is 26.6 m tall, not 25.0 m.
Both are measured from the horizontal
Everything in these two definitions hangs on one reference line: the horizontal through the observer's eye. Elevation turns upwards from it, depression turns downwards from it, and neither angle is ever measured from the vertical or from the ground under your feet.
This is where most lost marks come from. A student who measures from the vertical gets the complement of the correct angle, so a 32° elevation becomes 58°, and the answer that follows is wrong by a plausible-looking amount rather than obviously wrong. Drawing the horizontal dashed line before anything else costs five seconds and removes the error entirely.
Why the two angles in one diagram are equal
Picture a lighthouse keeper looking down at a boat and the boat's crew looking up at the keeper. There are two horizontal lines here — one at the top of the lighthouse, one at sea level — and they are parallel. The single line of sight cuts across both.
Alternate angles between parallel lines are equal, so the angle of depression from the lighthouse equals the angle of elevation from the boat. That is the fact worth carrying into an exam: a question phrased in terms of depression can be redrawn as an elevation angle inside the right-angled triangle you were going to use anyway, which puts the angle where you can actually label it.
The height the diagram quietly leaves out
Real measurements are taken from an instrument at eye level, not from the ground, and questions that mention a clinometer or a person's height are testing whether you noticed. The triangle you solve sits above the observer's eye, so its answer is the height above eye level, and the observer's height has to be added back at the end.
The same trap runs the other way for depression questions taken from a cliff or a window: the vertical side of the triangle is the height of the observation point above the object, which may not be the height of the cliff if the object is a boat sitting on the water rather than a point on the beach. Read what the vertical distance is actually joining before you write tan.
عام سوالات
Is the angle of depression the same as the angle of elevation?
They are different angles in different places, but they have the same size in a single diagram. The angle of depression from a tower down to a car equals the angle of elevation from that car up to the tower, because the two horizontal lines are parallel and the line of sight is a transversal cutting both.
Which trigonometric ratio should I use?
Label the triangle first, then choose. The horizontal distance is adjacent to the angle and the height is opposite it, so tangent handles most height problems. Sine or cosine are needed when the question gives or asks for the line of sight itself, which is the hypotenuse — for example the slant distance from an aircraft to a runway.
Why is my answer slightly different from the mark scheme?
Usually rounding. Work with the full calculator value throughout and round only the final answer, and check the calculator is in degree mode rather than radians. A tower height of 25.0 m becomes 0.44 m if the calculator is set to radians, which is a useful sanity check: if the answer is absurd, look at the mode.
Do angles of elevation appear at Foundation tier?
Right-angled trigonometry appears at both tiers on the current GCSE specifications, and elevation and depression questions are the standard way of dressing it up in context. Foundation questions tend to give a clean diagram with the angle already marked; Higher questions more often make you draw the diagram yourself from a written description.
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