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Download 13.1 Special Right Triangles
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Textbook: Chapter 13 ** Make sure that your calculator is set to the proper mode** Parts Of A Right Triangle Hypotenuse Acute Angles Leg Right Angle Leg The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. βNo man is free who cannot control himself.β β Pythagoras of Samos Pythagorean Theorem (leg)2 + (leg)2 = (hypotenuse)2 or a2 + b2 = c2 E1.) a = 3, b = 4 and c = ? 32 + 42 = π 2 π=5 E2.) a = ? , b = 36 and c = 39 π2 + 362 = 392 π = 15 E3.) a = 2 5, b = ? and c = 12 (2 5)2 +π2 = 122 π = 2 31 P1.) a = 5, b = ? and c = 13 P2.) a = 7, b = 3 and c = ? P3.) a = ?, b = 3 and c = 3 2 The sides of special right triangles (45-45-90 and 30-6090) have special relationships (ratios) Given one side of a certain right triangle, we can use these relationships (ratios) to unlock the other two sides. Why is this important if we already know the Pythagorean Theorem? We must know 2 sides of a right triangle to use the Pythagorean Theorem. 45-45-90 If you are given a: Leg (S) ο‘ The other leg is the same ο‘ Multiply by 2 to find the L (hypotenuse) Hypotenuse (L) ο‘ Divide by 2 to find the S (legs) b=5 c=5 2 a= 9 2 b=9 2 =9 a=2 6 b = 2 12 = 4 3 30-60-90 If you are given a: Small Leg (S) ο‘ ο‘ Multiply S by 3 to find M Multiply S by 2 to find L Medium Leg (M) ο‘ ο‘ Divide M by 3 to find S Multiply S by 2 to find L Hypotenuse (L) ο‘ ο‘ Divide L by 2 to find S Multiply S by 3 to find M *You need the S to unlock the M and L* b=5 3 c = 10 a=8 b=8 3 a= b= 10 = 3 20 3 3 10 3 3 Trigonometry Trigonometry β measurement of triangles Trigonometry Vocabulary -based on the acute angle (π) that is being used From β‘π΄ From β‘π΅ Opposite Hypotenuse Adjacent Adjacent Hypotenuse Opposite Trigonometric Ratio β a ratio of the lengths of two sides of a triangle. Sine (sin), Cosine (cos) and Tangent (tan) are the 3 basic trigonometric ratios Sin Cos Tan Solving Trigonometric Equations E1) Variable on top (multiply both sides by the denominator) π₯ 5 π₯ = 5πππ 32° π₯ β 4.24 πππ 32° = E2) Variable on bottom (flip-flop the trig function and the variable) 13 π ππ42° = π₯ 13 π₯= π ππ42° π₯ β 19.43 E3) Variable attached to the trigonometric function (inverse both sides) 12 13 12 ΞΈ = tanβ1 13 π β 42.71° π‘πππ = Solving Trigonometric Equations P1) Variable on top (multiply both sides by the denominator) sin 73 ° = π₯ 6 P2) Variable on bottom (flip-flop the trig function and the variable) tan 17 ° = 13 π₯ P3) Variable attached to the trigonometric function (inverse both sides) πππ π = 5 17 5 65 4 5 3 5 4 3 25 5 = 65 13 60 12 = 65 13 25 5 = 60 12 H O π₯ 15 π₯ = 15sin 29° π₯ β 7.27 π ππ 29° = O 13 π‘ππ 25° = π₯ 13 π₯= tan 25° π₯ β 27.88 π π π π πΆ π β β π A H A 7 13 7 Ξ = cos β1 13 π β 57.4° πππ π = π π π π πΆ π β β π Solving a right triangle means to find the measure of three angles of the triangle and three sides of the triangle. In other words all six parts. You can solve a right triangle if you know: (1) Two side lengths OR (2) One side length and one angle measure A= 41° B= 49° C= 90° a= 7.8 b= 9 c=11.9 Step 1: Find a missing side using the given information (Find c) B,b,c 9 sin 49° = π 9 π= sin 49° π β 11.9 Step 2: Find the other side (Find a) B,b,a 9 tan 49° = π 9 π= π‘ππ49° π β 7.8 A= B= C= 27°29β² 62°31β² 90° a= 6 b=11.5 c= 13 Step 1: Find the missing Side 62 + π 2 = 132 36 + π 2 = 169 π 2 = 133 π = 133 π β 11.5 Step 2: Find a missing angle using the given information (Find A) A,a,c 6 sin π΄ = 13 β1 6 A = sin 13 π΄ β 27°29β² B,a,c π 10 π = 10 cos 63° π β 4.5 ππππ‘ cos 63° = Angle of Elevation vs. Angle of Depression Angle of Elevation = Angle of Depression Angle of Elevation vs. Angle of Depression You are standing on top of a building that is 50 ft. tall and you see your buddy on the street. He is standing 30 ft. from the base of the building. Find the angle of depression between you and your buddy. 50 tan π₯ = 30 50 β1 π₯ = tan 30 π₯ β 59° The angle of elevation is 59° and because the angle of elevation is the same as the angle of depression, then the angle of depression is also 59°. π₯ 60 π₯ = 60 tan 65° π₯ β 128.7 ππ‘. tan 65° = π₯ 65° 60 ft Angle of Depression 200 ft π 175 ft 200 175 200 π = tanβ1 175 π β 48°49β² tan π =
