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Applications of Trig Functions • Today we’ll be looking at ways to solve realworld situations using the trig functions we learned last time. I like!! Sometimes we know the length of the sides of a triangle but need to know the angles. Guess what? We can use trig functions to do this! 7 x° 12 Get out your calculators we’re going to learn about Trig Inverses! The way of “undo-ing” trig functions! If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Use your calculator to find each angle measure to the nearest degree. A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71) cos-1(0.87) ≈ 30° sin-1(0.85) ≈ 58° tan-1(0.71) ≈ 35° Find m∠A. hypotenuse A 19 We should use SOH 13 SinA = ≈ .6842 19 SinA ≈ .6842 Sin −1SinA ≈ Sin −1 (.6842) € € C 13 B opposite Find m∠B. m∠B = 90° − m∠A A ≈ 43.17° m∠B = 90° − 43.17° = 46.83° € Find m∠B. hypotenuse A 23 We should use CAH 18 CosB = ≈ .7826 23 CosB ≈ .7826 Cos−1CosB ≈ Cos−1 (.7826) € € C 18 B adjacent Find m∠A. m∠A = 90° − m∠B B ≈ 38.5° m∠A = 90° − 38.5° = 51.5° € Find m∠B. A We should use TOA € 18 B adjacent TanB ≈ .5 Tan −1TanB ≈ Tan −1 (.5) € 9 C 9 TanB = ≈ .5 18 € opposite Find m∠B. m∠B = 90° − m∠A B ≈ 26.57° m∠B = 90° − 26.57° = 63.43° € Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. Remember that you can always use the Pythagorean Theorem too. Find ST, m∠R, and m∠T. RT2 = RS2 + ST2 (5.7)2 = 52 + ST2 Since the acute angles of a right triangle are complementary, m∠T ≈ 90° – 29° ≈ 61°. Find m∠D, EF, and DF Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? A 38% grade means the road rises (or falls) 38 ft for every 100 ft of horizontal distance. C 38 ft A 100 ft B An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Use the diagram above to classify each angle as an angle of elevation or angle of depression. The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70º, how tall is the Space Needle? Round to the nearest meter. Suppose a forest ranger sees a fire and measures the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. 3° An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance (x) between the two boats? Round to the nearest foot. A pilot flying at an altitude of 12,000 ft sights two towns directly in front of him. The angle of depression to one town is 78°, and the angle of depression to the second town is 19°. What is the distance (x) between the two towns? Round to the nearest foot. 19° 78° 12,000 ft 78° 19°