Angles of Elevation and Depression (VCE SSCE General Mathematics): Revision Notes

Worked Example: Finding Height Using Angle of Elevation

Problem: A park ranger measured the top of a plume of volcanic ash to be at an angle of elevation of $29°$. From her map she noted that the volcano was $8$ km away. Calculate the height of the plume to one decimal place.

Solution:

Step 1: Draw a right-angled triangle with the given information. Label the required height as $x$.

Step 2: Identify which trigonometric ratio to use. We know the adjacent side ($8$ km) and need to find the opposite side ($x$), so we use tangent.

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Step 3: Substitute the known values into the formula.

$\tan 29° = \frac{x}{8}$

Step 4: Multiply both sides by $8$ to solve for $x$.

$8 \times \tan 29° = x$

Step 5: Use a calculator to evaluate.

$x = 4.434...$

Step 6: Round to one decimal place.

$x = 4.4 \text{ km}$

Answer: The height of the ash plume is $4.4$ km.

Worked Example: Finding Distance Using Angle of Depression

Problem: From the top of a cliff that is $61$ m above sea level, Chen saw a capsized yacht. He estimated the angle of depression to be about $10°$. How far is the yacht from the base of the cliff, to the nearest metre?

Solution:

Step 1: Draw a diagram showing the given information. Label the required distance as $x$.

Step 2: Mark the angle inside the triangle. Due to alternate angles, the angle at the yacht corner is also $10°$.

Step 3: Identify which sides are involved. We have the opposite side ($61$ m) and need the adjacent side ($x$), so we use tangent.

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Step 4: Substitute the known values.

$\tan 10° = \frac{61}{x}$

Step 5: Multiply both sides by $x$.

$x \times \tan 10° = 61$

Step 6: Divide both sides by $\tan 10°$.

$x = \frac{61}{\tan 10°}$

Step 7: Calculate using a calculator.

$x = 345.948...$

Step 8: Round to the nearest metre.

$x = 346 \text{ m}$

Answer: The yacht is $346$ m from the base of the cliff.

Worked Example: Finding Height from Horizontal Distance

Problem: The angle of elevation of a rock climber scaling a vertical cliff is $47°$. The horizontal distance to the base of the cliff is $200$ m. How high is the climber up the face of the cliff? Answer to the nearest metre.

Solution:

Step 1: Identify the sides. The horizontal distance is the adjacent side ($200$ m), and we need to find the height (opposite side).

Step 2: Write the trigonometric equation using tangent.

$\tan 47° = \frac{\text{height}}{200}$

Step 3: Multiply both sides by $200$.

$\text{height} = 200 \times \tan 47°$

Step 4: Calculate.

$\text{height} = 214.45...$

Step 5: Round to the nearest metre.

$\text{height} = 214 \text{ m}$

Answer: The climber is $214$ m up the cliff face.

Worked Example: Multi-Step Problem with Two Triangles

Problem: A cable $100$ m long makes an angle of elevation of $41°$ with the top of a tower.

a) Find the height, $h$, of the tower to the nearest metre.

b) Find the angle of elevation, to the nearest degree, that a cable $200$ m long would make with the top of the tower.

Solution to part a:

Step 1: Focus on the triangle with the $100$ m cable. The opposite side is $h$ (height) and the hypotenuse is $100$ m.

Step 2: Use the sine ratio.

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Step 3: Substitute the known values.

$\sin 41° = \frac{h}{100}$

Step 4: Multiply both sides by $100$.

$h = 100 \times \sin 41°$

Step 5: Calculate and store the value.

$h = 65.6059...$

Step 6: Round to the nearest metre.

$h = 66 \text{ m}$

Answer: The height of the tower is $66$ m.

Solution to part b:

Step 1: Consider the triangle with the $200$ m cable. Let the angle be $\alpha$. The opposite side is $h$ and the hypotenuse is $200$ m.

Step 2: Use the sine ratio.

$\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}$

Step 3: Substitute the values (use the stored value of $h$ from part a).

$\sin \alpha = \frac{h}{200} = \frac{65.6059}{200}$

Step 4: Find $\alpha$ using the inverse sine function.

$\alpha = \sin^{-1}\left(\frac{65.6059}{200}\right)$

Step 5: Calculate.

$\alpha = 19.1492...°$

Step 6: Round to the nearest degree.

$\alpha = 19°$

Answer: The $200$ m cable makes an angle of elevation of $19°$ with the ground.