In the given diagram, Michael observes a buoy in the sea below him from a point six feet above a 45-foot cliff. He has been told that the perpendicular distance from the buoy to the base of the cliff is 60 feet. What is the angle of depression, in degrees, from Michael to the buoy? Give your solution to two decimal places.
So we have a diagram of Michael standing on top of a cliff looking out to sea. We can see then that we have a triangle formed by the horizontal, the vertical, and Michael’s line of sight down to the buoy. This triangle is right-angled as two of its sides are the horizontal and the vertical.
We’ve also been given the measurements of some of the sides for this triangle. We’re told that Michael is looking from a point six feet, presumably that’s his height, above a 45-foot cliff. Therefore, the length of the vertical side is the sum of those two values: 51 feet.
We were also told that the perpendicular distance from the buoy to the base of the cliff is 60 feet. So this is the length of the horizontal side of the triangle, side 𝐴𝐵. What we’ve been asked to do is calculate the angle of depression from Michael to the buoy.
Now remember, angles of depression are measured from the horizontal looking down at an object as we have in the diagram here. Therefore, the angle we need to calculate is the one I’ve marked here. And I’ve chosen to label it as angle 𝜃. Now let’s think about how we’re actually going to do this.
We have a right-angled triangle in which we know the lengths of two of the sides. And we want to calculate an angle. This suggests we need to use trigonometry in order to answer this problem. Let’s begin then by labelling the three sides of this triangle in relation to the angle 𝜃. So we have the opposite, the adjacent, and the hypotenuse.
We need to decide which of the three trigonometric ratios to use to calculate this angle. So let’s recall the acronym SOHCAHTOA to help with this. Remember S, C, and T stand for sine, cosine, and tan. And the O and the A and the H stand for opposite, adjacent, and hypotenuse.
In this question, the lengths we have been given are the lengths of the opposite and adjacent sides. So it’s the tan ratio that we’re going to use. So the definition of the tan ratio is that tan of an angle 𝜃 is equal to the opposite divided by the adjacent. Next, we’re going to substitute the values for the opposite and the adjacent sides in this question.
We have that tan of 𝜃 is equal to 51 over 60. Now we don’t just want to know the value of tan 𝜃, the ratio, we want to know the value of 𝜃 itself. So now we need to use the inverse tan function. This tells us that 𝜃 is equal to the inverse tan of the ratio 51 divided by 60.
Now at this point we can reach for our calculators in order to evaluate this. And it gives a value of 40.36453 and the decimal continues. Remember, the question has asked as to give the solution to two decimal places.
So finally, we need to round this value. So we have the angle of depression from Michael to the buoy to two decimal places is 40.36 degrees.