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P.o.D. β Consider triangle ABC with C=90 degrees. (on the board) 1.) If A=28 degrees and b=7, find c. 2.) If c=17 and a=15, find b. 3.) If B=45 degrees and c=23, find a. 4.) If A=76 degrees and a=1, find b. 5.) If a=6 and b=8, find c. 1.) 7.93 2.) 8 3.) 16.26 4.) 0.25 5.) 10 10-2: More Right Triangle Trigonometry Learning Target: I can determine the measure of an angle given its sine, cosine, or tangent; solve real-world problems using right triangle trig. EX: Consider a right triangle in which the hypotenuse is 19 and one leg is 15. What is the measure of the angle formed between these two sides? (draw a picture on the whiteboard) 15 cos π = β 19 15 ΞΈ = cos 19 β 37.86° β1 EX: Find the measures of all 3 angles in a triangle with side 3, 4, and 5. (draw a picture on the whiteboard) 4 tan π = β 3 4 β1 π = tan 3 = 53.13° 3 tan πΌ = β 4 3 β1 πΌ = tan 4 = 36.87° π½ = 180 β 53.13 β 36.87 = 90° EX: Find all 3 angles for the triangle below: b=18, c=33 B c=33 A b=18 C 18 cos π΄ = β 33 18 β1 π΄ = cos 33 β 56.94° 18 sin π΅ = β 33 18 β1 π΅ = sin β 33.06° 33 πΆ = 180 β 56.94 β 33.06 = 90° *Notice that angles A and B will always be complementary: 56.94+33.06=90. 3 EX: If cos π = , find sine and 4 tangent. *begin by drawing a triangle. A 4 π C B 3 *First, find the missing leg. 32 + π 2 = 42 β π 2 = 16 β 9 = 7 β π = β7 β7 sin π = 4 β7 tan π = 3 Letβs review the properties of a 30-60-90 triangle. A 60° 2x x 30° B C π₯β3 Leg opposite 30 = x Leg opposite 60 = π₯ β3 Hypotenuse = 2x EX: Find the sine, cosine, and tangent of 30 degrees. π₯ 1 sin 30° = = 2π₯ 2 π₯ β3 β3 cos 30° = = 2π₯ 2 π₯ tan 30° = = π₯ β3 1 = β3 1 β β3 β3 β3 = β3 3 EX: Find the sine, cosine, and tangent of 60 degrees. π₯ β3 β3 sin 60° = = 2π₯ 2 π₯ 1 cos 60° = = 2π₯ 2 π₯ β3 tan 60° = = β3 π₯ *Notice that sin 30° = cos 60° and sin 60° = cos 30°. These are known as cofunctions, because they are complements of one another. Letβs review the properties of a 45-45-90 triangle. 45 π₯β2 x 45 x EX: Find the sine, cosine, and tangent of 45 degrees. π₯ 1 β2 sin 45° = = = 2 π₯ β2 β2 π₯ 1 β2 cos 45° = = = 2 π₯ β2 β2 π₯ tan 45° = = 1 π₯ EX: A 115-ft ladder attached to a fire truck rests against the side of a building so that the top of the ladder is 111-ft above the ground. Find the angle formed by the truck and the ladder. 115 111 π 111 sin π = β 115 111 β1 π = sin β 115 74.84° Angle of Depression vs. Angle of Elevation (draw a picture on the whiteboard) EX: A hawk, cruising at an altitude of 215 feet, spots a prey at a direct distance of 406 feet. Find the angle of depression of the prey as seen by the hawk. hawk π 406 215 π 215 sin π = β 406 215 β1 π = sin β 406 31.98° *Remember, angle of depression is congruent to angle of elevation. Upon completion of this lesson, you should be able to: 1. Solve right triangles using trig functions. 2. Identify the properties of a 45/45/90 and 30/60/90 triangle. 3. Solve story problems involving right triangles. For more information, visit http://www.themathpage.com/atrig/solveright-triangles.htm HW Pg.673 2-9, 11-19