Flashcards · 11 cards

Elevation, depression & bearings

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Rule
Angles of elevation and depression are measured from what?
Answer
From the horizontal, up or down. The elevation from A to B equals the depression from B to A: they are alternate angles.
Elevation and depressionBoth from the horizontal, and equal: alternate angles
From the lesson: Angles of elevation and depression
Rule
An observer's eye is $e$ above the ground, $d$ from a building of height $H$ . The angle of elevation of the top?
Answer
$\tan\theta = \dfrac{H - e}{d}$ : take the eye height off first.
Eye heighttan θ = (H − e) ÷ d
From the lesson: Angles of elevation and depression
Rule
Two angles of elevation of the same top, from two points $d$ apart in a line: how do you find the height?
Answer
One height, two triangles: $h = x\tan\alpha = (x + d)\tan\beta$ . Solve for $x$ , then find $h$ .
Two triangles, one heighth = x tan α = (x + d) tan β
From the lesson: Angles of elevation and depression
Rule
A ladder of length $L$ makes an angle $\theta$ with the ground. How high does it reach, and how far out is its foot?
Answer
Height $L\sin\theta$ ; distance from the wall $L\cos\theta$ .
A ladder against a wallHeight L sin θ, distance out L cos θ
From the lesson: Angles of elevation and depression
Rule
How is a bearing measured?
Answer
From north, clockwise, with three figures: $065^\circ$ . "The bearing of B from A" is measured at A.
The bearing of B from ANorth line at A, then clockwise: 065°
From the lesson: Bearings
Rule
The bearing of B from A is $060^\circ$ . What is the bearing of A from B?
Answer
$060^\circ + 180^\circ = 240^\circ$ : a back bearing differs by $180^\circ$ .
Back bearingParallel north lines: 060° and 240° differ by 180°
From the lesson: Bearings
Rule
What do N $40^\circ$ E and S $30^\circ$ W mean as bearings?
Answer
N $40^\circ$ E: from north, $40^\circ$ towards east, so $040^\circ$ . S $30^\circ$ W: from south, $30^\circ$ towards west, so $210^\circ$ .
Compass directionsMeasured from north or south, towards east or west
From the lesson: Bearings
Rule
Going $d$ km on a bearing $\theta$ : how far north and how far east?
Answer
North $d\cos\theta$ , east $d\sin\theta$ .
North and east partsThe bearing is the angle at the start, next to the north side
From the lesson: Bearings
Rule
In a bearing diagram, how do you find the angle inside the triangle at a turning point $Q$ ?
Answer
Find the bearing from $Q$ back to the previous point (add or take $180^\circ$ ) and on to the next one. The angle is the difference.
The angle inside at QBack to P is 050° + 180° = 230°; on to R is 170°; the angle is 230° − 170° = 60°
From the lesson: Drawing the bearing diagram
Which method?
WAEC 2023 · Paper 1 · Q24
Mrs Kebeh stands at a distance of $110\text{ m}$ away from a building of vertical height $58\text{ m}$ . If Kebeh is $2\text{ m}$ tall, find the angle of elevation of the top of the building from her eye.
What height goes in the triangle?
Answer
The height above her eye: $58 - 2 = 56$ m. Then $\tan\theta = \frac{56}{110}$ .
From the lesson: Angles of elevation and depression Try the question
Which method?
WAEC 2021 · Paper 2 · Q3
The points $X$ , $Y$ and $Z$ are located such that $Y$ is $15\text{ km}$ south of $X$ , and $Z$ is $20\text{ km}$ from $X$ on a bearing of $270^\circ$ . Calculate, correct to:
two significant figures, $|YZ|$ ;
the nearest degree, the bearing of $Y$ from $Z$ .
What is the angle at $X$ ?
Answer
South and west ( $270^\circ$ ) are $90^\circ$ apart, so the triangle is right-angled at $X$ . Use Pythagoras, then find the bearing at $Z$ .
From the lesson: Drawing the bearing diagram Try the question