Lines of sight and angles

Looking up and looking down

Stand at the foot of a hill and look at the flag on a temple at the top. Your eyes follow one straight line to the flag. Now stand at the temple door and look down at a bus on the road. Again your eyes follow a straight line.

Trigonometry turns such pictures into right triangles. First we need the words.

• The line of sight is the straight line from your eye to the object you are looking at.
• The horizontal level is the flat line through your eye, parallel to the ground.
• The angle of elevation is the angle between the horizontal level and the line of sight when you look up at something above your eye.
• The angle of depression is the angle between the horizontal level and the line of sight when you look down at something below your eye.

Both angles are measured from the horizontal, never from the vertical. This is the first thing to check in every problem.

Why the two angles are equal

A climber C stands at the top of a cliff. A boat B floats at its foot. The line BC is one line of sight, seen from two ends.

The boat looks up at the climber, so the angle at B is an angle of elevation. The climber looks down at the boat, so the angle at C is an angle of depression.

The horizontal through B and the horizontal through C are parallel. The line of sight cuts both of them. So the two angles are alternate angles, and alternate angles between parallel lines are equal.

A worked example. From the top of a cliff, a climber sees a boat at an angle of depression of $38°$. Find the angle of elevation of the climber seen from the boat, and the angle between the line of sight and the cliff face at the top.

1. Equal angles. The angle of elevation at the boat is also $38°$.
2. The right triangle. The cliff meets the water at a right angle.
3. The angle at the top. The two acute angles add to $90°$, so the angle at the top is $90° - 38° = 52°$.

Move the sliders. The two angle readouts never differ, however tall the cliff is.

From a story to a right triangle

A word problem is a picture in disguise. Follow these steps every time.

1. Draw the ground and the vertical object. The tower, tree, pole or cliff stands at a right angle to the ground.
2. Draw the line of sight. It joins the eye to the top of the object, or the top to the thing seen below.
3. Mark the angle from the horizontal. Write the given angle at the correct corner.
4. Write the lengths. Put the known lengths and a letter for the unknown one on the triangle.
5. Choose the ratio. Look at the two sides you know and need.

The choice is quick once you name the sides from the marked angle.

• Height and ground distance: the height is opposite and the distance is adjacent. Use $\tan$.
• Height and slanting length, such as a ladder or a string: the height is opposite and the slanting length is the hypotenuse. Use $\sin$.
• Ground distance and slanting length: the distance is adjacent and the slanting length is the hypotenuse. Use $\cos$.

Values you will use

Most problems in this chapter use three angles. You met their ratios in the last chapter.

• $\sin 30° = \dfrac{1}{2}$, $\cos 30° = \dfrac{\sqrt{3}}{2}$ and $\tan 30° = \dfrac{1}{\sqrt{3}}$.
• $\sin 45° = \cos 45° = \dfrac{1}{\sqrt{2}}$ and $\tan 45° = 1$.
• $\sin 60° = \dfrac{\sqrt{3}}{2}$, $\cos 60° = \dfrac{1}{2}$ and $\tan 60° = \sqrt{3}$.

Use $\sqrt{3} \approx 1.73$ and $\sqrt{2} \approx 1.41$ when you need a number. Board papers say which values to take.

Practise with new numbers

Try it in Python

This program finds the angle of elevation of the top of three objects from a point on the ground. The angle of depression, seen from the top, is the same.

Common mistakes

• Measuring from the vertical. The angle of elevation or depression always starts at the horizontal line.
• Saying the two angles differ. For one line of sight they are equal, because they are alternate angles.
• Drawing the angle of depression at the object. It sits at the eye of the observer who looks down, between the horizontal and the line of sight.
• Skipping the diagram. Most wrong answers come from a triangle that was never drawn.

Remember these

• Line of sight: the straight line from the eye to the object.
• Elevation: looking up from the horizontal. Depression: looking down from the horizontal.
• The angle of elevation of the top equals the angle of depression of the bottom, for the same line of sight.
• Draw the right triangle first, name the sides from the marked angle, then choose $\sin$, $\cos$ or $\tan$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A bird sits on a wire above Dev's eye level. Which angle does Dev measure when he looks at it?

2. A lamp post is seen from the top of a building at an angle of depression of $27°$. What is the angle of elevation of the top of the building, seen from the lamp post?

3. The angle of elevation of the top of a tower is $40°$. What angle does the line of sight make with the tower at its top?

4. Bela finds the height of a tree and a ground distance. Which ratio should she use?

5. Ishan says the angle of depression of a boat is the angle between the line of sight and the cliff. Is he right?