Key Points

The line of sight is the straight line drawn from the eye of an observer to the point on the object being viewed. In diagrams, this line often represents the hypotenuse of a right-angled triangle.

The angle of elevation is the angle formed between the horizontal line from the observer's eye and the line of sight when the object is above the horizontal level. It is used when looking up.

The angle of depression is the angle formed between the horizontal line from the observer's eye and the line of sight when the object is below the horizontal level. It is used when looking down.

The angle of depression from an observer to an object is geometrically equal to the angle of elevation from that object back to the observer. This is due to them being alternate interior angles between two parallel horizontal lines.

To solve height and distance problems, first draw a clear diagram. Then, identify the right-angled triangle, label the known sides and angles, and use the appropriate trigonometric ratio ($\sin$, $\cos$, or $\tan$) to find the unknown quantity.

The tangent ratio, $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$, is most commonly used. It connects the height of an object (opposite side) and its horizontal distance from the observer (adjacent side).

The sine ratio, $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, is used when the problem involves the hypotenuse. For example, finding the length of a stretched rope or a ladder inclined at an angle.

The cosine ratio, $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, relates the horizontal distance (adjacent side) to the hypotenuse. It is used less frequently but is important when these two lengths are involved.

Memorize the values for standard angles $30^\circ$, $45^\circ$, and $60^\circ$. Key values are $\tan 30^\circ = \frac{1}{\sqrt{3}}$, $\tan 45^\circ = 1$, $\tan 60^\circ = \sqrt{3}$, and $\sin 30^\circ = \frac{1}{2}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

When the observer's height is given, the right-angled triangle is formed with the horizontal line at eye level. The final height of the object is the calculated vertical height from the triangle plus the height of the observer.

Problems involving two angles of elevation from one point (e.g., to the top and bottom of a statue on a pedestal) create two right triangles sharing a common base. Solve by setting up two equations, usually with the $\tan$ ratio.

Problems with angles from two different points along a line (e.g., walking towards a tower) create two right triangles with a common height. Set up two equations and solve them simultaneously to find the height and distances.