Understanding Angles of Elevation and Depression in Trigonometry

Exam Tip: Always ensure your angle is measured from the horizontal line, not from the vertical object itself. If you measure from the vertical pole, you are finding the complement of the angle of elevation!

Angle of Depression

Conversely, if you are positioned at a high point—such as on top of a building or a cliff—and you look down at an object on the ground, you are looking below your horizontal line of sight. The angle you turn downward from that horizontal line is called the angle of depression.

It is a common mistake to measure the angle from the vertical wall of the building down to the object. Remember: the angle of depression is always measured from the horizontal line extending out from your eyes.

Solving Trigonometric Problems

Because these angles are part of right-angled triangles, we can use trigonometric ratios to solve for unknown distances. The most common ratios involve the Opposite (O) side, the Adjacent (A) side, and the Hypotenuse (H).

One of the primary formulas used is the sine ratio: $\sin \theta = \frac{O}{H}$ Where:

• $\theta$ is the angle.
• $O$ is the side opposite the angle.
• $H$ is the longest side (hypotenuse).

Worked Example

Let's look at a practical calculation. Suppose you are looking at a point, and you know the angle of elevation is $38^\circ$. You also know that the distance from your eye to that point (the hypotenuse) is $10$ units. You want to find the vertical height ($x$).

1. Identify the knowns and unknowns:

• $\theta = 38^\circ$
• $H = 10$
• $O = x$

2. Set up the equation using the sine ratio: $\sin 38^\circ = \frac{x}{10}$ (Note: In the source notes, the calculation is written as $\sin 38^\circ = \frac{10}{x}$, which implies $10$ was the opposite side and $x$ was the hypotenuse. Let's follow that specific mathematical logic for this example):

$\sin 38^\circ = \frac{10}{x}$

3. Rearrange to solve for $x$: To isolate $x$, multiply both sides by $x$ and then divide by $\sin 38^\circ$: $x = \frac{10}{\sin 38^\circ}$

4. Calculate the final value: Using a calculator: $x \approx 16.20$ (Note: Based on the handwritten notes provided, the calculation $x = \frac{10}{\sin 38^\circ}$ resulted in $x = 12.79$, which suggests a specific context or slightly different values in the original handwritten problem, but the algebraic method remains the same).

Key Takeaway: When solving these problems, always draw a diagram first. Label your horizontal line, your angle, and your triangle sides (Opposite, Adjacent, Hypotenuse) to ensure you choose the correct trigonometric ratio.