Some Applications of Trigonometry
This chapter applies trigonometric ratios to find heights and distances of objects using the angle of elevation and angle of depression, without directly measuring them.
The angle of elevation is the angle formed between the horizontal line of sight and the line of sight to an object above it (looking up). The angle of depression is the angle formed between the horizontal and the line of sight to an object below it (looking down).
Such problems are generally solved by forming a right triangle using the height, the horizontal distance, and the line of sight as the hypotenuse, then applying the appropriate trigonometric ratio (usually tan) based on the given angle.
Q. A tower stands vertically on the ground. From a point 20 m away from the foot of the tower, the angle of elevation of the top is 60 degrees. Find the height of the tower.
Let the height be h. tan(60) = h/20, and tan(60) = root3. So h = 20 x root3 = 20root3 m (approximately 34.64 m).
The angle of elevation of an object as seen from a point is equal to the angle of depression of that same point as seen from the object, since both angles are measured from parallel horizontal lines, making them alternate angles.
Q. From the top of a 15 m high building, the angle of depression of a point on the ground is 30 degrees. Find the distance of the point from the base of the building.
Let the distance be d. tan(30) = 15/d, and tan(30) = 1/root3. So 1/root3 = 15/d, giving d = 15root3 m (approximately 25.98 m).
If the angle of elevation of the sun is 45 degrees, the length of the shadow of a pole is equal to:
Correct answer: (B) The height of the pole
When looking up at an object, the angle formed with the horizontal is called the angle of _______.
Elevation
Height and distance problems can always be solved without using any trigonometric ratios.
False — these problems specifically require trigonometric ratios (like sin, cos, tan) to relate the given angle to the unknown height or distance, since direct measurement is not possible or not given.