8-5 Angles of Elevation and Depression The Student will be able to: 1. Solve problems involving angles of elevation and depression. 2. Use angles of elevation and depression to find the distance between two objects.

Angles of Elevation and Depression Angle of elevation - the angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line. Angle of depression – the angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line. The two horizontal lines are parallel so the angle of elevation and the angle of depression are equal.

2nd – Which angle are you looking for? Example 1: The cross bar of a goalpost is 10 feet high. If a goal attempt is made 25 yards from the base of the goalpost that clears the goal by 1 foot, what is the smallest angle of elevation at which the ball could have been kicked to the nearest degree? 1 ft 1st – Draw a picture. 11 ft 2nd – Which angle are you looking for? 10 ft x° Where he kicked it from. 25 yds 75 ft 3rd – Which trig ratio applies and do you use it or its inverse? You’re given two sides. Use the inverse operation.

1st – Which side are we looking for? Example 2: A lifeguard is watching a beach from a line of sight 6 feet above the ground. She sees a swimmer at an angle of depression of 8°. How far away from the tower is the swimmer? 1st – Which side are we looking for? From the bottom of the tower to the swimmer. 6 ft 2nd – Which trig ratio applies and do you use it or its inverse? x° 8° You’re given one side and one angle. Use the trig ratio.

You Try It: 1. At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6 inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat’s head is 27°? 2. Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot? 60 ft 5.5 ft 25 ft 27° x 718.31 in or 59.86 ft ≈ x 31 ft 52° 40 ft 52° x 31.3 ft ≈ x

Two Angles of Elevation or Depression Angles of elevation or depression to two different objects can be used to estimate the distance between those objects. Similarly, the angles from two different positions of observation to the same object can be used to estimate the object’s height.

2nd – Which side are we looking for? Example 3: Two buildings are sited from atop a 200-meter skyscraper. Building A is sited at a 35° angle of depression, while Building B is sighted at a 36° angle of depression. How far apart are the two buildings to the nearest meter? 1st – Draw a picture. 35 36 2nd – Which side are we looking for? 200 The distance between the two buildings. 35 36 Hint: Find the length of the base of both triangles first. A x B

3rd – Which trig ratio applies and do you use it or its inverse? 35 3rd – Which trig ratio applies and do you use it or its inverse? 36 You’re given one side and one angle. You must solve for the base angle of each triangle. Find the base length of building b first. 200 36 35 A x B 275 4th – Find the distance between building a & building b. building a – building b = x 286 m – 275 m = x 11 m = x b = 275 m a = 286 m

1st – Which side are we looking for? You Try It: Miko and Tyler are visiting the Great Pyramid in Egypt. From where Miko is standing, the angle of elevation to the top of the pyramid is 48.6°. From Tyler’s position, the angle of elevation is 50°. If they are standing 20 feet apart, how tall is the pyramid? 1st – Which side are we looking for? The height of the pyramid. Hint: Find the base length of the smaller triangle first. y 3rd – Which trig ratio applies and do you use it or its inverse? y + 20 You’re given only one angle. You must solve for x.

4th – Solve for the bigger triangle. y Hint: Substitute for x from step 1. y + 20 5th – Substitute y back in original equation (step 1) and solve for x.