Raise your head

The idea

The angle of elevation of a point viewed is the angle formed by the line of sight with the horizontal when the point is above the horizontal level, as when we raise our head to look at it.

Look straight, then look up

The student looks straight ahead. Her line of sight is horizontal. Then she raises her head to look at the top of the Minar.

Now her line of sight makes an angle with the horizontal.

She walks closer to the Minar and keeps looking at its top. What happens to this angle?

• It gets bigger
• It gets smaller
• It stays the same

Walk to the angle

Drag the student along the ground, or tap the ground. The dashed line is the horizontal at her eye level. The purple arc is the angle she raises her head.

Up from the horizontal

The angle of elevation of a point viewed is the angle formed by the line of sight with the horizontal, when the point is above the horizontal level: the case when we raise our head to look at the object.

In the toy the angle grew as she walked closer. It is measured between the line of sight and the horizontal, not between the line of sight and the Minar.

Notes

The angle of elevation is the angle formed by the line of sight with the horizontal when the point viewed is above the horizontal level, as when we raise our head.

Check yourself

Which angle is the angle of elevation of the top C of the Minar from the eye A of the student?

In the right triangle ABC (right angle at B), the angle ACB at the top of the Minar is 40°. What is the angle of elevation ∠BAC, in degrees?

Answer: 50

∠BAC = 180° − 90° − 40° = 50°.

The student walks away from the Minar and keeps looking at its top. What happens to the angle of elevation?

For which point do we measure an angle of elevation?

• ∠ACB. The angle at C lies between the line of sight and the Minar. It is not measured from the horizontal.
• ∠BAC — correct. Yes! It lies between the line of sight AC and the horizontal AB.
• ∠CAE. AE goes straight down to her feet, so ∠CAE does not start from the horizontal.
• It gets bigger.. Walking away makes the line of sight flatter. You saw this in the toy: the angle gets smaller.
• It stays the same.. The angle changes with her distance from the Minar.
• It gets smaller. — correct. Yes! The farther she stands, the flatter her line of sight, so the angle of elevation is smaller.
• A fish below the surface of a river. We lower our head to look at it. That is an angle of depression.
• A kite high above the horizontal level — correct. Yes! We raise our head to look at it, so the point is above the horizontal level.
• A point exactly at the level of our eyes. The line of sight is the horizontal itself, so there is no angle to measure.