1. Quiz 1 Need-to-Know Arithmetic Mean (AM) or average: (a + b) / 2 Geometric Mean (GM): √ab Altitude = GM of divided hypotenuse Pythagorean Theorem: a 2 + b 2 = c 2 Pythagorean Triples: Whole numbers that solve the theorem 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 a b alt = √ab alt 45 6 3√2 6 3 3√3 30 60

2. 5-Minute Check on Lesson 7-3 Transparency 7-4 Click the mouse button or press the Space Bar to display the answers. Find x and y. 1. 2. 3. The length of a diagonal of a square is 15√2 cm. Find the perimeter of the square. 4. The side of an equilateral triangle measures 21 inches. Find the length of the altitude of the triangle. 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. In the right triangle find CD if DE = 5.? Standardized Test Practice: ACBD 55√3(5/3)√3 10 x y°y° x = 16 y = 16√3 P = 60 cm (-3,3) B x = 5√2 y = 45° 30° y x 10.5√3 ≈ 18.19 in 32 CD E 3x° 6x°

3. Lesson 7-4a Right Triangle Trigonometry

4. Trigonometric Functions Main Trig Functions: –Sinesin-1 ≤ range ≤ 1 –Cosinecos-1 ≤ range ≤ 1 –Tangenttan-∞ ≤ range ≤ ∞ Others: –Cosecantcsc 1 / sin –Secantsec 1 / cos –Cotangentcot 1/ tan –Tangent sin / cos

5. Trig Definitions Sin (angle) = Cos (angle) = Tan (angle) = Opposite ---------------- Hypotenuse Adjacent ---------------- Hypotenuse Opposite ---------------- Adjacent S-O-HS-O-H C-A-HC-A-H T-O-AT-O-A

6. Ways to Remember S-O-H C-A-H T-O-A Some Old Hillbilly Caught Another Hillbilly Throwing Old Apples Some Old Hippie Caught Another Hippie Tripping On Acid Extra-credit: Your saying

7. θ hypotenuse A B C Example: opposite side BC sin A = sin θ = ---------------------- = ------ hypotenuse AB Use trig functions to help find a missing side in a right triangle. Format: some side Trig Function ( an angle, θ for example) = ----------------------- 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

8. θ hypotenuse A B C Use inverse trig functions to help find a missing angle in a right ∆. Format: some side Trig Function -1 (-------------------------) = missing angle, θ for example some other side where the trig function -1 is found using 2 nd key then the trig function on calculator Example: opposite side BC sin A = sin θ = ---------------------- = ------ hypotenuse AB If BC = 7 and AB = 14, then to find  A or θ we have opposite side BC 7 sin θ = ---------------------- = ----- = ----- = 0.5  A = θ = sin -1 (0.5) = 30° hypotenuse AB 14 Anatomy of a Trig Function

9. 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:

10. 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:

11. Example 3 Use a calculator to find tan to the nearest ten thousandth. KEYSTROKES: 56 1.482560969 TANENTER Answer: KEYSTROKES: 90 0 COSENTER Answer: Use a calculator to find cos to the nearest ten thousandth.

12. a. Use a calculator to find sin 48° to the nearest ten thousandth. b. Use a calculator to find cos 85° to the nearest ten thousandth. Example 4 Answer:

13. 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