Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 1 of 8

Looking up and looking down: which ratio solves the triangle?

A lookout sights a boat and the boat's crew sight the lookout. This lesson compares the two angles, then chooses between sine, cosine and tangent, and between multiplying and dividing.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a right-angled triangle has an angle of 40° and an opposite side of 7 cm. Which ratio links the opposite side to the hypotenuse?
    AnswerSine: sin 40° = opposite ÷ hypotenuse.
  2. From Year 9: two parallel lines are cut by a transversal. One alternate angle is 37°. What is the other?
    Answer37°. Alternate angles between parallel lines are equal.
  3. From Year 10: round 53.468 to one decimal place.
    Answer53.5, because the digit after the first decimal place is 6.
Words used in this lesson
angle of elevation
the angle measured up from a level line to a line of sight. For example: looking up from a harbour to a hilltop.
angle of depression
the angle measured down from a level line to a line of sight. For example: a bird looking down at a fish.
line of sight
the straight line from the eye to the thing being looked at. For example: a laser pointer's beam.
horizontal
level, parallel to the ground. For example: the surface of still water.
opposite
the side across the triangle from the marked angle. For example: the side not touching the angle.
inverse tangent
the key that finds an angle from a tangent ratio. For example: tan−1 on the calculator.

Two people, one sight line

A guide stands on a lookout above a lake and measures the angle between the level horizon and her line of sight down to a boat. That is the angle of depression. Aboard the boat, the captain measures the angle between the water and his line of sight up to the lookout. That is the angle of elevation.

Predict first: is the captain's angle bigger than the guide's, smaller, or the same size? Write a reason in words, then use the first figure to test it.

30
25
64.3 mhorizontal distance to the boat
25.0°angle of elevation at the boat

Height 30 m, depression 25°: distance 30 ÷ tan 25° ≈ 64.3 m. Seen from the boat, the elevation is 25.0°.

Show me the steps
  1. Height 30, depression 25°: the distance is 30 ÷ tan 25° ≈ 64.3 m. The angle at the boat, found from height and distance, is 25.0°: the same.
  2. Make the height 50 and keep 25°: the boat is further out, 50 ÷ tan 25° ≈ 107.2 m, and the elevation at the boat is still 25.0°.
  3. Make the height 30 and the angle 60°: the boat is close in, 30 ÷ tan 60° ≈ 17.3 m, and the elevation is 60.0°.
  4. Make the height 10 and the angle 20°: 10 ÷ tan 20° ≈ 27.5 m. Whatever the height and the angle, the two angles match, so one measurement gives both.

Depression down from the horizontal at the top equals elevation up from the horizontal at the bottom, because the two horizontal lines are parallel and the angles are alternate. Both angles are measured from a level line, never from the vertical.

Choosing the ratio: draw the right-angled triangle, mark the angle, name the sides opposite, adjacent and hypotenuse, and pick the ratio that uses the known side and the wanted side. Then, if the wanted side is the top of the fraction you multiply the known side by the ratio; if it is the bottom you divide.

1
40
sin, multiplyratio and operation
6.43wanted length

Known: hypotenuse 10. Wanted: opposite. Use sin, and multiply: opposite = 10 × sin 40° ≈ 6.43.

Show me the steps
  1. Known the hypotenuse (10), wanted the opposite, θ = 40°: the two sides are opposite and hypotenuse, so sin. The wanted side is the top of the fraction: opposite = 10 × sin 40° ≈ 6.43.
  2. Pair 4 (known the opposite 10, wanted the hypotenuse): the same two sides, so the same ratio sin, but now the wanted side is at the bottom: hypotenuse = 10 ÷ sin 40° ≈ 15.56.
  3. Pair 3 (known the adjacent, wanted the opposite) with θ = 55°: the sides are opposite and adjacent, so tan. Opposite = 10 × tan 55° ≈ 14.28.
  4. Pair 6 (known the adjacent, wanted the hypotenuse) with θ = 65°: adjacent and hypotenuse, so cos. Hypotenuse = 10 ÷ cos 65° ≈ 23.66. A divide gives a longer side than the one known.

Depression is not measured from the vertical. A depression of 12° is a shallow glance, only a little below level. If you put 12° between the cliff face and the sight line, the triangle has the wrong angle and every answer after that is wrong too.

Worked examples

1. From a point on level ground 30 m from the foot of a spire, the angle of elevation of the top is 52°. How tall is the spire, to 1 decimal place?

  1. The 30 m is next to the angle (adjacent) and the height is opposite it, so use tan.
  2. tan 52° = h ÷ 30, so h = 30 × tan 52° = 38.4 m.

2. A rowing boat is seen from a bridge deck 22 m above the water at an angle of depression of 17°. How far is it from the foot of the bridge, to 1 decimal place?

  1. The angle of elevation at the boat is also 17°. The 22 m is opposite it and the wanted distance is adjacent: tan.
  2. tan 17° = 22 ÷ d, so d = 22 ÷ tan 17° = 72.0 m. The distance is longer than the height, as it should be for a small angle.

3. A zip line falls 14 m over a horizontal distance of 60 m. What is its angle of depression, to 1 decimal place?

  1. Opposite 14, adjacent 60: tan θ = 14 ÷ 60.
  2. θ = tan−1(14 ÷ 60) = 13.1°.

Your go. The first step is given. From a point on the ground, the angle of elevation of the top of a 25 m mast is 38°. How far is the point from the foot of the mast, to 1 decimal place?

  1. The 25 m is opposite the 38° angle, and the distance is adjacent, so tan 38° = 25 ÷ d.
  2. Rearrange for d: is it 25 × tan 38° or 25 ÷ tan 38°?
Check the answer

d = 25 ÷ tan 38° = 32.0 m. Multiplying instead, 25 × tan 38° = 19.5, would put the point closer to the mast than the mast is tall, which cannot happen for an angle smaller than 45°.

With a GDC (optional)

Check the angle mode first: the display must be in degrees (look for DEG or D in the mode or set-up menu). To find an angle, use the inverse key above the tan, sin or cos key, often marked tan−1, and type the ratio as a fraction in brackets. Enter a whole calculation once and round only at the end.

Before pressing anything, say whether the answer should be bigger or smaller than the side you know. Everything here can be done with a scientific calculator, and the working on the page is the same either way.

Check before you go on

1. From a window 15 m above the ground, a car is seen at an angle of depression of 32°. Which calculation gives the horizontal distance from the building to the car?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. From a point on level ground 45 m from the foot of a tower, the angle of elevation of the top is 28°. How tall is the tower, to 1 decimal place?

3. A ramp rises 1.2 m over a horizontal distance of 9 m. What is its angle of elevation, to 1 decimal place?

Level 2 · Building

4. A lighthouse is 36 m tall. From its top, the angle of depression of a boat is 24°. How far is the boat from the foot of the lighthouse, to the nearest metre?

5. A surveyor in a boat sees the top of a 50 m cliff at an angle of elevation of 21°. How long is the straight line of sight from the boat to the top, to 1 decimal place?

6. From a point 70 m from a building, the angle of elevation of the bottom of a flagpole on the roof is 32° and of the top is 41°. How tall is the flagpole, to 1 decimal place?

Level 3 · Stretch

7. From a cliff top 80 m above the sea, two boats are in line with the foot of the cliff. Their angles of depression are 15° and 40°. How far apart are the boats, to the nearest metre?

8. A surveyor sights the top of a tower at an angle of elevation of 20°. She walks 40 m straight towards the tower and now sees the top at 35°. How tall is the tower, to 1 decimal place?

Something to think about

A lookout's angle of depression to a boat is 25°. The boat sails away until the angle of depression is 10°. Does the boat's distance more than double, or less? Test it with the first figure, keeping the height fixed.

One way to think about it

More than double: at 25° the distance is h ÷ tan 25° ≈ 2.14h and at 10° it is h ÷ tan 10° ≈ 5.67h. The angle fell by a factor of 2.5 but the distance grew by a factor of about 2.6, because 1 ÷ tan grows faster than the angle shrinks near zero.

Exit ticket

9. A kite's string is 85 m long and makes an angle of 36° with level ground. Taking the string as straight, how high is the kite?