Some Applications of Trigonometry

Line of sight

The line from an observer's eye to the point on the object being looked at. Angles of elevation and depression are measured between this line and the horizontal.

Angle of elevation

The angle the line of sight makes with the horizontal when you look up at something above your eye level, as when you raise your head to see the top of a tower.

Angle of depression

The angle between the line of sight and the horizontal when the point being viewed is below the observer's horizontal level, as when you look down from a balcony at something on the ground.

Relation between the two angles: why does the angle of depression from P to B equal the angle of elevation of P from B?

The horizontal through P and the horizontal through B are parallel, and the line PB cuts across both. The two angles are alternate angles on that transversal, so they are equal.

Choosing the ratio: how do you decide which trigonometric ratio to use in a heights-and-distances problem?

Pick the ratio that links the side you know with the side you want, relative to the known angle. Height and horizontal distance are the two legs, so use tan or cot; if a slant length such as a ladder is involved, use sin or cos.

Observer's height: why must it be added when finding the height of a tall object?

The angle of elevation is measured from the observer's eye, not from the ground. The trigonometric ratio gives only the height above eye level, so the observer's height is added; a 1.5 m observer seeing 28.5 m above eye level gives a 30 m chimney.

Heights and distances: how do you set up a problem of this kind?

Draw a simple figure and mark the right triangle or triangles it contains, with the known angle, the known length and the unknown. Choose the ratio that connects them, solve for the unknown, and add or subtract any extra lengths such as an observer's height.

From a point 15 m from the foot of a tower, the angle of elevation of its top is 60°. How do you find the tower's height?

The height and the 15 m distance are the two legs of a right triangle, so use tan 60° = height ÷ 15. Since tan 60° = √3, the height is 15√3 m.

An electrician must reach 3.7 m up a pole using a ladder at 60° to the ground. How do you find the ladder's length and the distance of its foot from the pole?

The ladder is the hypotenuse, so sin 60° = 3.7 ÷ ladder, giving about 4.28 m. The foot's distance is the adjacent side, so it equals 3.7 × cot 60° = 3.7/√3, about 2.14 m.

A 10 m building's top is seen at 30° and the top of a flagstaff on it at 45°, from the same point P. How do you find the flagstaff's length?

From the building alone, tan 30° = 10 ÷ AP, so AP = 10√3 m. Let the flagstaff be x; then tan 45° = (10 + x) ÷ 10√3, so 10 + x = 10√3 and x = 10(√3 − 1), about 7.32 m.

A tower's shadow is 40 m longer when the Sun's altitude is 30° than when it is 60°. How do you find the tower's height?

Let the height be h and the shorter shadow x. From the 60° triangle, h = x√3; from the 30° triangle, h ÷ (x + 40) = 1/√3. Substituting gives 3x = x + 40, so x = 20 and h = 20√3 m.

From the top of a tall building, the top and bottom of an 8 m building are seen at depressions of 30° and 45°. How do you find the tall building's height?

Turn each depression into an equal elevation using alternate angles. The 45° angle makes the tall building's height equal the gap between them. With PD the height above the smaller roof, the 30° triangle gives PD + 8 = PD√3, so PD = 4(√3 + 1) and the height is 4(3 + √3) m.

From a point on a bridge 3 m above the banks, the banks are seen at depressions of 30° and 45°. How do you find the river's width?

Drop a perpendicular from the point to the river line and split the width into two parts. The 30° side gives tan 30° = 3 ÷ AD, so AD = 3√3 m; the 45° side gives BD = 3 m. The width is 3 + 3√3 = 3(1 + √3) m.

From a lighthouse of height h, two ships on the same side are seen at depressions of 30° and 45°. How do you find the distance between them?

Convert each depression into an equal angle of elevation at the ship. The nearer ship at 45° is h/tan 45° = h away, and the farther ship at 30° is h/tan 30° = h√3 away. Their distance apart is the difference, h(√3 − 1).

The top of a building is seen from the foot of a tower at one angle, and the tower's top from the building's foot at another. How do you find the building's height from the tower's?

Both angles use the same ground distance between the two feet. Find that distance from the triangle whose height you know, the tower's, using its angle. Then use it with the other angle's tangent to get the building's height.

Why does the angle of elevation of a building's top increase as you walk towards it?

The building's height above your eye stays the same while your horizontal distance shrinks. So tan of the angle, height divided by distance, gets larger, and a larger tangent means a larger acute angle, as when it rises from 30° to 60°.

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