Right Triangle Trigonometry Answer Key

Honors Algebra Trig Unit 10 Name: KEY Block: 13.1 Right Triangle Trigonometry Goal: to solve for the sides and angles of triangles Triangles consist of two components: 3 Sides & 3 angles Note: rather than using x as our unknown when dealing with angles theta A is often used. Trigonometry comes from the Greek words meaning triangle measurement. A RIGHT TRIANGLE has 1 right angle. The other 2 angles are acute because they are less than 90 GIVEN: Two Sides of a Right triangle Pythagorean Theorem: a ^ 2 + b ^ 2 = c ^ 2 Only works with Right triangles A ab are legs C is the hypotenuse Chypotenuse b leg a leg B **The hypotenuse across from the right angle and is is always the longest side.** Practice: Find the measure of the third/missing side of the following triangles. Leave answers as radicals if necessary. NO DECIMALS! 1. 4 3 9 + x ^ 2 = 16 3 ^ 2 + x ^ 2 = 4 ^ 2 x ^ 2 = 7 12 2. 9 81+| 44 = x ^ 2 9 ^ 2 + 12 ^ 2 = x ^ 2 225 = x ^ 2 X = 15 3. 16 12 16 ^ 2 + 12 ^ 2 = x ^ 2 400 = x ^ 2 256 + 144 = x ^ 2 10 4. 5. 5 6. 13 5 5 ^ 2 + x ^ 2 = 13 ^ 2 25 + x ^ 2 = 169 x ^ 2 = 144 5 ^ 2 + 5 ^ 2 = x ^ 2 5 + 25 = x ^ 2 50 = x ^ 2 X = 12 x = 5sqrt(2) 4 ^ 2 + x ^ 2 = 10 ^ 2 16 + x ^ 2 = 100 x ^ 2 = 84 x = 2sqrt(21) X = 20 x = sqrt(7) GIVEN: One Side of a Right triangle and a Special Case Angle: 45", 30", or 60" Special Right Triangles: There are two special right triangles. Case 1: 45°-45°-90°: In a 45°-45°-90° A, the hypotenuse is sqrt(2) times as long as a leg. *A 45°-45°-90° A is a(n) (sosceles A, so the legs are equal formula: hyp leg.12 45 x12 45 X Case 2: 30°-60°-90°: In a 30 deg - 60 deg -90^ triangle, the hypotenuse is 2 times as long as the short and the longer leg is sqrt(3) times as long as the short "Remember that the shortest side is across from the smallest angle. leg. leg, formulas: hyp = short leg.2 long leg = Short leg 13 30 2x hypotenuse long * MUST find Short Leg 1 ^ S^ + if it isn't given! leg 60° short leg X Practice: Use special right triangles to determine the length of the missing sides. Leave your answers exact-NO decimals!!! 7. 30 12 8. 5sqrt(2) 45 9. 7/(- sqrt(3)) = (2/3)/3 (14sqrt(3))/3 6sqrt(3) 10 5 5 7 10. 45 10sqrt(2) 10 11. 6sqrt(3) 30 12 6 12. 412 8 4sqrt(2) 45 6 8/(sqrt(2)) * (sqrt(2))/(sqrt(2)) = (8sqrt(2))/2 = 4sqrt(2) GIVEN: One Side of a Right triangle and a Non-Special Angle: (NOT 45", 30", or 60") • Two Sides of a Right triangle and you need to find an angle Trigonometry: There are six trig ratios that come from the ratios of the sides of a right triangle. These 6 ratios are: sine sin cosine Cos We will work only with the first 3. The sides will be referred to as: tangent tan secant sec cosecant CSC cotangent cot opposite adjacent hypotenuse hyp hyp opp adj 0 adj opp The ratios formed by the sides have names/labels: Remember Soh Cah Toa tan 0-opp adj sin - OPP hyp cos - adj hyp Write 3 basic trigonometric ratios for the following triangle. sin 0= 3 hyp cos tan 0- Opp 4 B A 13 C sin B = 3/6 = boxed 1 2 cos B = (3sqrt(3))/6 = boxed -3 2 tan B = 3/(3sqrt(3)) = 1/(sqrt(3)) = boxed -3 3 Use a calculator to complete, CALC MUST be in DEGREE mode! 16) sin 549 x = 93in * 54 deg x = 7.28 15) tan 43 deg = 0.933 18)25tan 35 deg = x/25 * 25 13) cos 22 deg = 0.927 14) sin 79°982 17) 5000s 20^ * = x/52 <= 2 x = 48.86 x = 52cos 20 deg 19) K = 3/(500 ^ (- 10 ^ 2)) x = 3, 19 sin 70 deg = 3/x 20) cos 54 14 X COS54 x = 23.8 14 82 21) tan 54 X x = 5 ^ 9 * 1.58 X = 82/(tan 54 deg) x = 25tan 35 x = 17.51 sin A = (3sqrt(3))/6 = boxed - sqrt 3 2 tan A - (root(3, 2))/3 = boxed sqrt 3 cos A = 3/6 = boxed 1 2 3sqrt(3) Using Trig to Solve for missing sides Steps: 1. Label your sides according to the angle provided 2. Use SOH-CAH-70A to pick which trig, function to use 3. Set up your ratio and solve-IMPORTANTI a. le adjacent, hypotenuse, opposite a. Using one krown side and angle and one unknown alde You need to set up the ratio to see how to solve for the missing side b. You cannot solve these problems mentally it needs to be written downl Find the value of x & y to the nearest tenth. Show the trig ratio. 22. x = 43.9 23. x = 2.5 x hup 14 OPP 10 27 o*d_{j} 0s * 52 deg = 27/x tan 10 ^ 8 = x/14 x = 14 + an * 10 deg Using Inverse Trig to Solve for missing angles COS Solve for .x. 25.) sin x^ * = 2/7 x = arcsin(2/7) = boxed 16.6 24. y = 9.5 x = 3.1 hyp 10 18 Y Opp x adj cos 72 ^ 2 = x/10 x = 10cos 72 deg All three inverse trig, functions are found on your calculator. Press 2nd → and then select the trig. function you want to use! 10 26.) cos x 17 x = arccos(10/17) = boxed 54^ Examples: Solve for the missing angle using the two sides that are given: 28) cos theta = 5/10 theta = arccos(binomial(5,3) * or) arccos(1/2) 29.) tan theta = 3/17 theta = arctan(3/17) 10 3 OPP sadj 17 adj 10 deg y =10sin72^ prime prime sin 72 deg = y/10 x = arctan((sqrt(2))/5) = boxed 15.8 27.)tan x^ * = (sqrt(2))/5 30.) sin theta = 12.5/52 theta = arcsin(12.5/52) 52 hyp 12.5 opp x = 27/(cos 52 deg) 13.9 ^ 3 sin^ - 1 tan^ - 1 52 deg 72 deg