trigonometry question

Question

The straight-line distance of an airplane to a control tower is 36.2 km. If the angle of elevation from the tower to the plane is 4°, and the plane is 24 km south of the tower, how much west of the tower is the plane?

Answer 1

Hi Isaac,

This is a classic angle of elevation question. Let's first make our strategy by starting with the givens.

We have two cardinal directions in the problem: south and west. So the first diagram I would sketch is a birds-eye view of the ground with the x and y axes. Let these distances represent the ground distance from tower to plane (rather than the distance from the ground into the sky to the plane).

Let the x axis represent west to east.

Let the y axis represent north to south.

For simplicity, let the origin (0,0) be the location of the tower.

Next, we sketch the plane's location in relation to the tower.

Where is the plane?

If it's 24 km south of the tower, then we draw a point on the negative y-axis between the 3rd and 4th quadrants. Label the line down the y-axis as 24 km.

The plane is to the west of the tower, so we sketch a dot indicating the plane in the 3rd quadrant.

A new diagonal line from the plane's dot to the tower (origin) creates a hypotenuse of a right triangle. The x distance from the y axis to the plane's dot is the base side length of the triangle and the answer to the question "How much west of the tower is the plane?"

This is the first triangle showing ground distances.

We also need a second triangle involving the angle of elevation:

The key to angle of elev (or angle of depression) questions is sketching a simple right triangle with a horizontal line as the ground. The angle pointing up (elevation) or down is always from that horizontal line.

Example of the 4º angle of elevation from the tower pointing toward the plane:

Plane -----_____

| '''''''''-------_____

| ''''''''-------_____Tower

---------------------------------------------------------------------- (ground)

"a" (ground distance)

Draw a straight line between the plan and tower. This is the hypotenuse of this second right triangle.

That line is the given value of 36.2 km. Note, this is different than the hypot. of the first triangle because that represents the ground distance. 36.2 km is the distance from the ground tower into the air to the plane.

Let the angle at the tower from the ground to the hypotenuse be 4º.

Suppose we label the distance along the ground as "a" for adjacent.

• Solve for a using a trig function, the angle 4º, and the hypotenuse.

"a" is the ground distance.

What line does this correspond to in the first triangle drawn on the x-y plane?

• Label the hypot. of the first triangle as "a" and use the value from the trig function solution above.

• Using the first triangle side lengths of "a" and 24km, what theorem do you use to find the distance from the y axis to the plane? That distance is the answer to the question.

Solutions:

"a" = 36.11 km using cos(4º) = ground distance from tower to plane.

Distance from west of tower to plane = 27 km by the Pythagorean theorem.

You got this.