Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
SANTA CLARA COUNTY OFFICE OF EDUCATION North County Regional Occupational Program (NCROP) Engineering Technology Trigonometry Tutorial & Problems c Θ2 a Θ1 b Characteristics of the Right Triangle One of the three angles is 90º The sum of the three angles is 180º. (θ1º+θ2º+90º = 180º) The two sides are length a and b The long side, or hypotenuse, is length c Pythagorean's Theorem: In a right triangle whose legs have length a and b, and whose hypotenuse has length c, then c2 = a2 + b2. Example 1: If side a = 5 inches and side b = 10 inches, what is the length of hypotenuse c? Pythagorean's Theorem states c2 = a2 + b2 c2 = (5 inches)2 + (10 inches)2 c2 = 25 inches2 + 100 inches2 c2 = 125 inches2 c = √(125 inches2) This is an acceptable answer c = √(5*5*5 inches2) c = 5√5 inches This is a better answer c = 11.18 Use a calculator for the best answer Example 2: If side a = 6 inches and hypotenuse c = 12 inches, what is the length of side b? Pythagorean's Theorem states c2 = a2 + b2 c2 – a2 = b2 (12 inches)2 – (6 inches)2 = b2 144 inches2 – 36 inches2 = b2 108 inches2 = b2 √(108 inches2) = b This is an acceptable answer 2 √(2*2*3*3*3 inches ) = b 6√3 inches = b This is a better answer 10.39 = b Use a calculator for the best answer SANTA CLARA COUNTY OFFICE OF EDUCATION North County Regional Occupational Program (NCROP) Engineering Technology c = 5 inches Θ2 a = 3 inches Θ1 b = 4 inches Sine (abbreviated 'sin'): In a right triangle, sin of an angle θ is the length of the opposite side divided by the length of the hypotenuse. Example: If side a = 3 inches, side b = 4 inches, and side c = 5 inches, what is sinθ1 and sinθ2? sinθ1 = (side a / hypotenuse c) sinθ1 = (3 inches / 5 inches) sinθ1 = 0.60 sinθ2 = (side b / hypotenuse c) sinθ2 = (4 inches / 5 inches) sinθ2 = 0.80 Cosine (abbreviated 'cos'): In a right triangle, cos of an angle θ is the length of the adjacent side divided by the length of the hypotenuse. cosθ1 = (side b / hypotenuse c) cosθ1 = (4 inches / 5 inches) cosθ1 = 0.80 cosθ2 = (side b / hypotenuse c) cosθ2 = (3 inches / 5 inches) cosθ2 = 0.60 A common way to memorize these relationships is the term, ‘SOHCAHTOA’ ‘SOH” represents ‘Sine is Opposite over Hypotenuse’ ‘CAH’ represents ‘Cosine is Adjacent over Hypotenuse’ ‘TOA’ represents ‘Tangent is Opposite over Adjacent’ SANTA CLARA COUNTY OFFICE OF EDUCATION North County Regional Occupational Program (NCROP) Engineering Technology c a b Given the right triangle above and the following lengths, calculate the length of the missing side. (Hint: Don't forget the units) Problem 1 2 3 4 5 6 7 8 9 10 Length of Side a 3 meters 5 yards 1 inch 8 miles 5 feet 8 inches 1 mm Length of Side b 4 meters 8 yards 1 inch 4 miles 1 foot 8 mm 2 yards Length of Hypotenuse c 3 feet 15 mm 3 yards 13 feet 16 inches 3 mm SANTA CLARA COUNTY OFFICE OF EDUCATION North County Regional Occupational Program (NCROP) Engineering Technology c Θ2 a Θ1 b Given the right triangle above, calculate the sin and cos given the following edge lengths. The first one is done as an example. # Ex Length a 4 miles Length b 6 miles Hypotenuse c 2√13 miles 5 feet Problem a) sinθ1= _________ sinθ1 = a/c =(4 miles) / (2√13 miles) =2/√13 sinθ1= Problem b) cosθ1= __________ cosθ1 = b/c =(6 miles) / (2√13 miles) =3/√13 sinθ2= 1 3 feet 4 feet 2 5 meters 12 meters 13 meters cosθ1= sinθ1= 3 6 inches 4 inches 2√13 inches cosθ1= cosθ2= 4 1 mm 3√7 mm 8 mm sinθ2= cosθ2= 5 2√21 cm 4 cm 10 cm sinθ1= cosθ2=