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Wednesday: Warm-up • Draw a unit circle and label all the key angles in degrees. • You also need a calculator for today! 1 2 Section 4-3 Right Triangle Trigonometry Objectives • I can use Special Triangle Rules • I can use SOH-CAH-TOA to find information from right triangles and word problems 4 Special Right Triangles 30o, 45o, 60o 60o You must memorize these!!! 1 2 3 2 3 2 The x-value is the cosine of that angle. 2 2 2 2 2 2 The y-value is the sine of that angle. Use the pythagorean theorem to find the sides. 1 2 The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: hyp the side opposite the acute angle , opp the side adjacent to the acute angle , θ and the hypotenuse of the right triangle. adj Memory Aide: SOH-CAH-TOA sine, cosine, tangent, cotangent, secant, and cosecant. opp sin = cos = adj tan = opp hyp hyp adj csc = hyp opp sec = hyp adj cot = adj opp 6 Calculate the trigonometric functions for . 5 4 3 The six trig ratios are 4 sin = 5 4 tan = 3 5 sec = 3 3 cos = 5 3 cot = 4 5 csc = 4 7 Calculator Mode • MUST be set to DEGREES!! 8 Finding an Angle 5 2 We have the opposite side and hypotenuse Sin θ = 2/5 = sin-1(2/5) = 23.58° 9 Word Problems • Always draw a picture or diagram to represent the situation. 10 Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation. line of sight object angle of elevation horizontal observer When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: horizontal angle of depression line of sight object observer angle of depression. 11 Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff? observer cliff 42 m horizontal 16○ angle of depression line of sight 16○ d 42 42 tan16 d= = 146.47. d tan 16 The ship is 146.47 m from the base of the cliff. ship 12 Example 3: A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder? ladder house 14 16 sin = 14 16 θ Next use the inverse sine function to find . = sin1(14/16) = 61.044975 The angle formed by the ladder and the ground is 61.04 The painter’s plan is unsafe! 13 Homework • WS 8-1 • Study Special Triangle values 14