Trigonometry Lesson 7.4

What is Trigonometry? Trigonometry is the study of how the sides and angles of a triangle are related to each other. It's all about triangles!

What is a Trigonometric Ratio? A ratio of the lengths of the sides of a right triangle. The three most common trigonometric Ratios are: –S–Sine  “sin” –C–Cosine  “cos” –T–Tangent  “tan”

Trig Definitions: S-O-H-C-A-H-T-O-A 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

Ways to Remember S-O-H C-A-H T-O-A Some Old Hillbilly Caught Another Hillbilly Throwing Old Apples She Offered Her Cat A Heaping Teaspoon Of Acid Extra-credit opportunity: Your own saying

θ θ Key Concepts of Trigonometric Ratios 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

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

SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle sin L = = 0.47 cos L = = 0.88 tan L = = 0.53 sin N = = 0.88 cos N = = 0.47 tan N = = 1.88 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 rounded to nearest hundredth. Example 1 L M N

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 = 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

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

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° Cos Tan Sin

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

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

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.

How to Find an Angle In trigonometry, you can find the measure of the angle by using the inverse of sine, cosine, and tangent. Sin (angle) = x  angle = sin -1 (x) Cos (angle) = y  angle = cos -1 (y) Tan (angle) = z  angle = tan -1 (z)

SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle 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 6: Example 6

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° x 60° 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

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° x 45° x°x°

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) = x 16 54° 7. x°x°

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 # odd (15 problems)