A security light is being installed outside a loading dock.  The light is mounted 20 feet above the ground.  The light must be placed at an angle so that it will illuminate the end of the parking lot.  If the end of the parking lot is 100 feet from the loading dock, what should be the angle of depression of the light?  What information do we have in order to hang the light at the correct angle?  We will answer these questions later in the lesson.

In a previous lesson we used the unit circle to determine the value of trigonometric functions using that radius of 1.

This chart lists some frequently used values.

Sometimes you know a trigonometric ratio, but not the angle itself.  In this case, you need to use an 

 of the function.  The inverse of a sine function is the 

Example:      sin θ = √3 / 2         can be written as           θ =  arcsin √3 / 2

which you can then read as θ is any angle whose sine value is √3 /2.  This same rationale can be applied to other inverses, leading to the 

You can use the provided table to solve equations like sin θ = √3 / 2, arriving at the inverse equation θ = arcsin √3 / 2 and finding that θ = 60°, 120°, or any angle that is coterminal.  Using this technique, 🔑a) find the measure(s) of θ when cos θ = -√2 / 2.

What if you are asked to find the value of cos (arcsin 2/3)?  Here, it helps to draw a diagram first.  Then, you can use the “formula” from earlier and state that angle θ = arcsin 2/3.  Therefore, sin θ = 2/3 by the definition of an inverse.  This means that in your diagram, θ should be your central angle and the sides would be opposite = 2, and hypotenuse = 3.  Use the Pythagorean theorem and find the adjacent side, which is found to = √5.  Negative roots are irrelevant because we are talking about a positive length of a side.  Then, using your third side, the cos θ is √5 / 3, so cos (arcsin 2/3) = √5 / 3.

🔑Try these examples, assume the angles are in Quadrant I.

b)cos (arccos 4/5)             c) tan (arccos 3/5)         d) cos (arcsin 2/5)

If you have some side lengths of a right triangle, but not the measurement of an angle, you can use this inverse relationship to find the angle.  🔑e) Draw a right triangle DEF, where angle D is the right angle.  Now, using lowercase letters, label the sides of the triangle - opposite sides receive the label of the opposite angles.  Now, set up an equation to find out what cosine of angle E is.  Solve for the angle measurement if side f = 17 and d = 32.  Round your answer to the nearest tenth, and use a calculator.

Ah, now, how about the security light!?  If you remember, the light is to be mounted 20 feet off of the ground, and reach 100 feet across a parking lot.  🔑f) What is the angle of depression for the light’s beam of focus?  Use the inverse relations that we have discussed to solve!

Speaking of “solve!”, you can use information from previous lessons and this lesson to completely solve a right triangle.  To 

, you need to find the measure of all sides and angles, and usually only need two pieces of information and a calculator to do so.  Example:  Given - ΔABC is a right triangle, m∠A = 33° and b = 5.8.

🔑g)Draw a diagram where ∠C = 90°.  If A is 33, the third angle, ∠B, must be 57°.  We can also find the three side lengths using the tan and cos of given ∠A.  What are the lengths of the sides and angles of ΔABC?

Use the same triangle diagram.  Solve the triangle for each of the following given information.  You can round to the nearest tenth in your answers.

🔑Answer the questions below.  Round to the nearest tenth when necessary.

n) The steepest railway in the world is the Katoomba Scenic Railway in Australia.  the passenger car is pulled up the mountain by twin steel cables on a track of 1020 feet to obtain a change in altitude of 647 feet.  Find the angle of elevation of the railway and how far the car travels over the ground.

o) The rim of a basketball hoop is 10 feet above the ground.  The free throw line is 15 feet from the basket.  If the eyes of a basketball player are 6 feet off of the ground, what is the angle of elevation of the player’s line of sight when shooting a free throw?

p) Years ago, a section on I-75 near Cincinnati, Ohio had a rise of 8 meters per 100 meters of horizontal distance.  However, there were numerous accidents involving large trucks on this section of highway.  Engineers decided to reconstruct the highway so that there is only a rise of 5 meters per 100 meters of horizontal distance.  Find the original angle of elevation and the new angle of elevation.

q) Review:  A traffic helicopter is flying 1000 feet above the downtown area of a city.  To the right, the pilot sees the baseball stadium at an angle of depression of 63°.  To the left, the pilot sees the football stadium at an angle of depression of 18°.  What is the distance between the two stadiums?

r)🦄Completely solve the given triangle.

Students will discover how to find inverses of trigonometric functions and solve applications to find angle measures.

Arcs and Angles