Apply sine and cosine rules to find distances or angles of elevation

For the same triangle:

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

When to use the cosine rule:

• When we know two sides and the included angle (SAS)
• When we know all three sides (SSS)

The angle of elevation is the angle formed by the horizontal line from the observer's eye and the line of sight to an object above the observer.

In practical problems, we often combine trigonometry (using sine and cosine rules) with angle of elevation to find distances that are difficult to measure directly.

A student standing at point A on the ground observes the top of a radio tower at point C. The angle of elevation to the top of the tower is 35°. The student then walks 50 meters directly away from the tower to point B. From point B, the angle of elevation to the top of the tower is 20°. Find the height of the tower (point C to the ground).

Solution

Step 1: Draw a diagram and identify the triangle

Let:

• h = height of the tower (in meters)
• AB = 50 m (distance the student walked)
• Let the distance from A to the base of the tower be x meters

Then from point B, the distance to the base of the tower is (x + 50) meters.

Step 2: Set up equations using tangent

In triangle ACD (from point A): $\tan 35° = \frac{h}{x}$

So: $h = x \tan 35°$ ...(1)

In triangle BCD (from point B): $\tan 20° = \frac{h}{x + 50}$

So: $h = (x + 50) \tan 20°$ ...(2)

Step 3: Equate the two expressions for h

$x \tan 35° = (x + 50) \tan 20°$

$x \tan 35° = x \tan 20° + 50 \tan 20°$

$x (\tan 35° - \tan 20°) = 50 \tan 20°$

$x = \frac{50 \tan 20°}{\tan 35° - \tan 20°}$

Calculating values:

• tan 20° ≈ 0.3640
• tan 35° ≈ 0.7002

$x = \frac{50 \times 0.3640}{0.7002 - 0.3640} = \frac{18.2}{0.3362} \approx 54.1 \text{ m}$

Step 4: Find the height of the tower

From equation (1): $h = x \tan 35° = 54.1 \times 0.7002 \approx 37.9 \text{ m}$

Therefore, the height of the radio tower is approximately 38 meters.

• Use the sine rule when you have two angles and one side, or two sides and a non-included angle
• Use the cosine rule when you have two sides and the included angle, or all three sides
• For angle of elevation problems, draw a clear diagram and use trigonometric ratios (tan, sin, cos) along with the sine or cosine rules as needed

In Tanzania, surveyors and engineers use sine and cosine rules when planning road construction in mountainous areas like the Highlands region. For example, when building the new road from Dar es Salaam to Makambako, engineers must calculate distances across valleys and angles of elevation to determine safe gradients for vehicles. Similarly, telecommunications companies apply these rules to position mobile phone towers, ensuring proper coverage by calculating angles of elevation from different observation points to determine the optimal tower height.