Solve a practical right-angled-triangle problem

A cable reaches a point 8 m above level ground and makes an angle θ above horizontal. Which expression gives its length? The sketch is not to scale

A model keeps the horizontal run fixed at 5 m while increasing the angle above horizontal. Predict the required straight rail length

Change the angle for a fixed 5 m horizontal run. Observe the rise and the length of rail required for each slope

Which equation models the required rail length L?

Model the measured endpoints

Write what each length measures before choosing the ratio. Keep lengths in a common unit, and distinguish a height above ground from a rise above an instrument

1. 1Unknown sloping length When horizontal and vertical legs form the right angle, the sloping cable or rail is the hypotenuse. Divide the known opposite or adjacent leg by the matching sine or cosine
2. 2Unknown ground distance With a known vertical rise and an angle above horizontal, tangent is rise divided by ground distance \(\displaystyle d=\frac{\text{rise}}{\tan\theta}\)
3. 3Raised observation point On level ground, subtract the instrument height from the total target height before using the rise \(\displaystyle \text{rise}=H-h_0\)

Model

\(\displaystyle L=\frac{8}{\sin52^{\circ}}\)

A cable attaches 8 m above level ground. Its angle above horizontal is 52°. Find its length to one decimal place. The sketch is not to scale

Model

\(\displaystyle L=\frac{5.5}{\cos34^{\circ}}\)

A straight rail has a horizontal run of 5.5 m and makes 34° above horizontal. Find its length to one decimal place. The sketch is not to scale

Model

\(\displaystyle d=\frac{14}{\tan39^{\circ}}\)

A vertical mast is 14 m high on level ground. Its top is seen at 39° above horizontal from a ground-level point. Find the horizontal distance to one decimal place. The sketch is not to scale

Model

\(\displaystyle d=\frac{14-1.6}{\tan39^{\circ}}\)

A 1.6 m high instrument sees the top of the same 14 m mast at 39° above horizontal. Both stand on level ground. Find the horizontal distance to one decimal place. The sketch is not to scale

Model

\(\displaystyle L=\frac{2.8}{\sin43^{\circ}}\)

A brace makes 43° with a vertical wall and reaches 280 cm horizontally from its base. Find the sloping brace length in metres, to two decimal places. The sketch is not to scale

A 1.6 m high instrument views the top of a 14 m mast at 39° above horizontal. Both stand on level ground. A student uses the calculation below for the horizontal distance. What is wrong?