Lesson 16: Applications of Trig Ratios to Find Missing Angles

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1 : Applications of Trig Ratios to Find Missing Angles Learning Targets I can find a missing angle in a right triangle diagram and apply this to real world situation Opening Exercise Find the shadow cast by a 10 foot lamp post when the angle of elevation of the sun is 58. Find the length to the nearest tenth of a foot. Remember Basic Trigonometric Functions (SOH CAH TOA) We can use the basic trigonometric functions to find the measure of an acute angle of a right triangle if you know two of the sides of the triangle. To find the actual number of degrees in the angle we will use "inverse function" They appear as sin -1, cos -1 and tan -1 or arcsine, arccosine, and arctangent Example 1. Use the inverse functions on your calculator to evaluate the following. Round your answers to the nearest tenth of a degree. a) cos (x) = m x = cos 1 (0.5431) m x b) tan (x) = m x c) sin (θ) = m θ d) tan (B) = 3.5 m B e) sin (x) = m x

2 Example 2. Step 1. Choose the correct trig function and set up and equation substituting the given lengths Step 2. Use the inverse function on the calculator to find the angle (remember the calculator needs to be on degree mode) Example 3. Use the inverse functions on your calculator to find the missing angles. Round your answers to the nearest tenth of a degree. a) b) c) d)

3 Try on your own Example 4. Angle of Elevation and Depression The angle of elevation or depression is the angle between the horizontal (parallel with the Earth s surface) and the line of sight. Example 5. Use BUCK$ to read and solve the question Suppose a tree 50 feet in height casts a shadow of length 60 feet. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground?

4 Example 6. An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport? Example 7. A man standing on level ground is 1000 feet away from the base of a 350-foot-tall building. Find, to the nearest degree, the measure of the angle of elevation to the top of the building from the point on the ground where the man is standing.

5 : Applications of Trig Ratios to find Missing Angles Classwork. Use BUCK$ to understand and solve each problem 1. Find the value of x. Round your answers to the nearest tenth of a degree a) b) c) d) 2. Ron and Francine are building a ramp for performing skateboard stunts, as shown in the accompanying diagram. The ramp is 7 feet long and 3 feet high. What is the measure of the angle, x, that the ramp makes with the ground, to the nearest tenth of a degree? 3. A group of friends have hiked to the top of the Mile High Mountain. When they look down, they can see their campsite, which they know is approximately 3 miles from the base of the mountain. What is the angle of depression?

6 4. A roller coaster travels 80 ft of track from the loading zone before reaching its peak. The horizontal distance between the loading zone and the base of the peak is 50 ft. At what angle is the roller coaster rising according to the model? 5. A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram. To the nearest degree, what is the value of x? 6. Gwen has built and raised a wall of her new house. To keep the wall standing upright while she builds the next wall, she supports the wall with a brace, as shown in the diagram below. What is value of p, the measure of the angle formed by the brace and the wall? Brace p Wall 7. A communications company is building a 30-foot antenna to carry cell phone transmissions. As shown in the diagram below, a 50-foot wire from the top of the antenna to the ground is used to stabilize the antenna. Find, to the nearest degree, the measure of the angle that the wire makes with the ground.

7 : Applications of Trig Ratios to find Missing Angles Homework Use BUCK$ to read and solve each problem 1. For each triangle shown, use the given information to find the indicated angle to the nearest degree a) b) 2. A 12 meter flagpole casts a 9 meter shadow. Find the angle of elevation of the sun. 3. A fire department s longest ladder is 110 feet long, and the safety regulation states that they can use it for rescues up to 100 feet off the ground. What is the maximum safe angle of elevation for the rescue ladder?

8 4. The tallest television transmitting tower in the world is in North Dakota, and it is 2059 feet tall. If you are on level ground exactly 5280 feet (one mile) from the base of the tower, what is your angle of elevation looking up at the top of the tower? 5. In the accompanying diagram, the base of a 15-foot ladder rests on the ground 4 feet from a 6-foot fence. a If the ladder touches the top of the fence and the side of a building, what angle, to the nearest degree, does the ladder make with the ground? b Using the angle found in part a, determine how far the top of the ladder reaches up the side of the building, to the nearest foot.