1. 5-Minute Check on Lesson 7-4a Transparency 7-5a Click the mouse button or press the Space Bar to display the answers. Find x. 1. 2. 3. Given an adjacent side and the hypotenuse, which trig function do you use? 4. Given an opposite side and the hypotenuse, which trig function do you use? 5. ∆MNP is a 45°- 45°- 90° triangle with right angle P. Find NP if MN = 20. 6. In the right triangle which trig function would you use to find CD with  C? Standardized Test Practice: ACBD cossintan sec 6 x°x° 10 x ≈ 17.43 cos 10√2 ≈ 14.14 C x ≈ 53.14° 33° x sin 32 CD E 37° 5

2. Lesson 7-4b Right Triangle Trigonometry

3. 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

4. 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

5. θ θ Crash Trig Course What’s Constant: Side opposite right angle is the hypotenuse What Changes: Side opposite the angle, θ; Side adjacent to the angle, θ In the triangle to the left: AC is opposite of θ and BC is adjacent to it In the triangle to the right: AC is adjacent to θ and BC is opposite it A B C hypotenuse A B C Left-most Triangle: opposite side AC sin θ = ------------------- = ------ hypotenuse AB adjacent side BC cos θ = ------------------- = ------ hypotenuse AB opposite side AC tan θ = ------------------- = ------ adjacent side BC Right-most Triangle: opposite side BC sin θ = ------------------- = ------ hypotenuse AB adjacent side AC cos θ = ------------------- = ------ hypotenuse AB opposite side BC tan θ = ------------------- = ------ adjacent side AC opposite adjacent opposite adjacent

6. Steps to Solve Trig Problems Step 1: Draw a triangle to fit problem Step 2: Label sides from angle’s view –H: hypotenuse –O: opposite –A: adjacent Step 3: Identify trig function to use –Circle what values you have or are looking for –SOH CAH TOA Step 4: Set up equation Step 5: Solve for variable

7. Remember: Sin 90° is 1 Cos 90° is 0 Tan 90° is undefined SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle sin 2 θ + cos 2 θ = 1 (from Pythagorean Theorem) x°x° 12 8 tan x° = 12/8 x = tan -1 (12/8) x = 56.31° When looking for an angle use the inverse of the appropriate trig function (2 nd key then trig function on your calculator) 12 is opposite the angle x; and 8 is adjacent to it: opp, adj  use tangent Example 1: Example 1

8. Remember: Sin 90° is 1 Cos 90° is 0 Tan 90° is undefined SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle x 17 52° tan 52° = 17/x x tan 52° = 17 x = 17/tan 52° x = 13.28 When looking for a side use the appropriate trig function (based on your angle and its relationship to x, and your given side). 17 is opposite of the angle and x is adjacent to it: opp, adj  use tangent Example 2: Example 2

9. Remember: Sin 90° is 1 Cos 90° is 0 Tan 90° is undefined SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle x 47° 13 sin 47° = x/13 13 sin 47° = x 9.51 = x 13 is the hypotenuse (opposite from the 90 degree angle) and x is opposite from given angle: opp, hyp  use sin Example 3: Example 3

10. Check Yourself You have a hypotenuse and an adjacent side Use: _______Solve: x = ___ You have an opposite and adjacent side Use: _______Solve: y = ___ You have an opposite side and a hypotenuse Use: _______Solve: z = ___ 35° 15 y 35° z x 25 55° 21.42 14.34 26.15 Cos Tan Sin

11. Example 4 EXERCISING A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches. Step 1: Draw a triangle to fit problem Step 2: Label sides from angle’s view Step 3: Identify trig function to use Step 4: Set up equation Step 5: Solve for variable Hyp Opp S  O / H C  A / H T  O / A Opp y sin 7° = -------- = ----- Hyp 60

12. Example 4 cont KEYSTROKES: 60 7 7.312160604 SINENTER Multiply each side by 60. Use a calculator to find y. Answer: The treadmill is about 7.3 inches high.

13. Example 5 CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, how high does the ramp rise off the ground to the nearest inch? Answer: about 15 in.

14. Side, x opposite 30° and 24 is the hyp  sin 30 = x/24 x = 24 sin 30 = 12 1) Identify what you are trying to find (variable) – Side or Angle 2) Relate given (opp, adj, hyp, angle) to the variable 3) Solve for variable x 24 30° 1. x°x° 20 15 2. x 60° 30 3. Side, x adjacent 60 and 30 is the hyp  cos 60 = x/30 x = 30 cos 60 = 15 Angle, x opposite 20 leg and 15 is adj leg  tan x = 20/15 x = tan -1 (20/15) = 53.1 Trig Practice

15. 1) Identify what you are trying to find (variable) – Side or Angle 2) Relate given (opp, adj, hyp, angle) to the variable 3) Solve for variable Side, x opposite 49 and 13 is the hyp  sin 49 = x/13 x = 13 sin 49 = 9.81 Side, x is hypotenuse and 12 is adj leg  cos 45 = 12/x x = 12/(cos 45) = 12√2 Angle, x is opposite 12 and 18 is hyp  sin x = 12/18 x = cos -1 (12/18) = 48.2 Trig Practice cont x 49° 13 4. x 45° 12 5. x°x° 18 12 6.

16. 1) Identify what you are trying to find (variable) – Side or Angle 2) Relate given (opp, adj, hyp, angle) to the variable 3) Solve for variable Trig Practice cont Angle, x is opposite 12 and adj to 10  tan x = 12/10 x = tan -1 (12/10) = 50.2 Side, x is adjacent 54 and 16 is opp  tan 54 = 16/x x = 16/(tan 54) = 11.62 x 16 54° 7. x°x° 12 10 8.

17. Summary & Homework Summary: –Trigonometric ratios can be used to find measures in right triangles –Identify what you are trying to find (variable) – Side or Angle –Relate given (opp, adj, hyp, angle) to the variable –Solve for variable Homework: –pg 368, 18-21, 43-46, 61