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Quiz 1 Need-to-Know Arithmetic Mean (AM) or average: (a + b) / 2 Geometric Mean (GM): √ab a Altitude = GM of divided hypotenuse b alt = √ab Pythagorean Theorem: a2 + b2 = c2 Pythagorean Triples: Whole numbers that solve the theorem 45 6 Side opposite 30° angle is ½ the hypotenuse Side opposite 45° angle is ½ the hypotenuse times √2 Side opposite 60° angle is ½ the hypotenuse times √3 6 45 3√2 30 60 3 Transparency 7-4 5-Minute Check on Lesson 7-3 Find x and y. 1. x = 16 32 x = 5√2 2. x 10 y = 16√3 30° y° x y = 45° y 3. The length of a diagonal of a square is 15√2 cm. Find the perimeter of the square. P = 60 cm 4. The side of an equilateral triangle measures 21 inches. Find the length of the altitude of the triangle. 10.5√3 ≈ 18.19 in 5. ∆MNP is a 45°- 45°- 90° triangle with right angle P. Find the coordinates of M in quadrant II with P(2,3) and N(2,8). 6. Standardized Test Practice: In the right triangle (-3,3) C D 3x° find CD if DE = 5.? 6x° A 5 B 5√3 C (5/3)√3 Click the mouse button or press the Space Bar to display the answers. D 10 E Lesson 7-4a Right Triangle Trigonometry Trigonometric Functions • Main Trig Functions: – Sine – Cosine – Tangent sin cos tan -1 ≤ range ≤ 1 -1 ≤ range ≤ 1 -∞ ≤ range ≤ ∞ • Others: – Cosecant – Secant – Cotangent – Tangent csc sec cot 1 / sin 1 / cos 1/ tan sin / cos Trig Definitions • Sin (angle) = Opposite ---------------Hypotenuse S-O-H • Cos (angle) = Adjacent ---------------Hypotenuse C-A-H • Tan (angle) = Opposite ---------------Adjacent T-O-A Ways to Remember • S-O-H Some Old Hillbilly Caught Another Hillbilly Throwing Old Apples • C-A-H • T-O-A Some Old Hippie Caught Another Hippie Tripping On Acid Extra-credit: Your saying Anatomy of a Trig Function A Example: θ opposite side BC sin A = sin θ = ---------------------- = -----hypotenuse AB C B Use trig functions to help find a missing side in a right triangle. Format: Trig Function ( an angle, θ for example) = some side ----------------------some other side where the some side or the some other side is the missing side If θ = 30 and AB = 14, then to find BC we have opposite side BC BC sin θ = sin 30 = 0.5 = ---------------------- = ----- = -----hypotenuse AB 14 (14) 0.5 = BC = 7 Anatomy of a Trig Function A Example: θ opposite side BC sin A = sin θ = ---------------------- = -----hypotenuse AB C Use inverse trig functions to help find a missing angle in a right ∆. Format: Trig Function -1 B some side (-------------------------) = missing angle, θ for example some other side where the trig function -1 is found using 2nd key then the trig function on calculator If BC = 7 and AB = 14, then to find A or θ we have opposite side BC 7 sin θ = ---------------------- = ----- = ----- = 0.5 hypotenuse AB 14 A = θ = sin-1(0.5) = 30° Example 1 Find sin L, cos L, tan L, sin N, cos N, and tan N. Express each ratio as a fraction and as a decimal. Answer: Example 2 Find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal. Answer: Example 3 Use a calculator to find tan thousandth. KEYSTROKES: TAN 56 to the nearest ten ENTER 1.482560969 Answer: Use a calculator to find cos thousandth. KEYSTROKES: COS Answer: 90 to the nearest ten ENTER 0 Example 4 a. Use a calculator to find sin 48° to the nearest ten thousandth. Answer: b. Use a calculator to find cos 85° to the nearest ten thousandth. Answer: Summary & Homework • Summary: – Trigonometric ratios can be used to find measures in right triangles – Sin of an angle is opposite / hypotenuse – Cos of an angle is adjacent / hypotenuse – Tan of an angle is adjacent / hypotenuse • Homework: – pg 367-368; 1, 4, 5-8, 11, 15, 16