MTH 122: Section 204. Plane Trigonometry. Test 1

Transcription

1 MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π 3π/2 2π cos α sin α tan α b). Evaluate i). sin 1 ( 3/2) ii). cos 1 ( 2/2) iii). tan 1 ( 3/3) 2. a). Sketch angle α = 490 b). Convert 270 to radian. c). Convert 7π 2 to degrees. 3. Find the exact value of the following. a). cos(7π/3) b). tan(315 ) c). sin(2π/3) 1

2 4. Using reference angle, match questions (a), (b) and (c) with either of (I-VIII). Simply put no match if there is no match a) cos(514 ) b). cos(291 ) c) cos( 480 ) I) cos(69 ) II) cos(69 ) III). cos(26 ) IV). cos(26 ) V). cos(60 ) VI). cos(60 ) VII). cos(111 ) VIII). cos(111 ). 5. Without using calculator, find the exact value of sin α, cos α, tan α, csc α, sec α and cot α, where α is an angle in the standard position whose terminal side contains the point P( 3, 1). i). sin α ii). cos α iii). tan α iv). csc α v). sec α vi). cot α 2

3 Section B: The use of calculator is allowed. Show your work and clearly identify your answer. 6. List ALL angles α between 0 and 360 that satisfy the given equation. a). cos α = 3 2 b). sin α = 2 2 c). tan α = a). Find the length of the arc intercepted by central angle θ = 45 in a circle of radius 1.5 f t. b). Given that the area of the sector of an angle intercepted by central angle θ in a circle of radius r = 2 f t is 10π f t 2. Find the central angle θ in degrees. 8. If cos α = 4 7 and α is in Quadrant III, evaluate i). sin α ii). tan α iii). sec α iv). csc α v). cot α. 9. Given the triangle 5 α 7 β find a). sin α b). cos α c). tan α d). sin β e). cos β f). tan β g). α h). β. 3

4 10. a). Convert to degrees and round to 4 decimal places. b). Convert to degree-minute-seconds D M S. c). Simplify and give answer in D M S i) ii) iii) a). Sketch the function y = 2 cos(x + π/4) + 3 b). Identify the i). domain ii). range iii). phase shift iv). period v). amplitude vi). x-intercepts 4

5 MTH 122 Plane Trigonometry Test 2. Answer all questions. Name (1) By comparing the curve given below with curve y = sin x or y = cos x, describe all properties listed below and use your result to write the equation of the curve in the form y = A sin (B [x C])+ D or y = A cos (B [x C]) + D a. Domain: b. Range: c. Phase shift: d. Amplitude e. Period: f. Write as y = A sin (B [x C]) + D or y = A cos (B [x C]) + D. (2) Without using calculator, simplify as a single trigonometric value or find exact value of i). sin 12 o cos 33 o + cos 12 o sin 33 o ii). cos 57 o cos 33 o sin 57 o sin 33 o iii). sin(3π/12) + sin(π/12) iv). 8 tan 15 o 1 tan 2 15 o v). sin(52.5 ) sin(7.5 ). 1

6 (3)Verify the following. State (with valid reason(s)) if TRUE or FALSE. State the name of identity used. a). (cos α tan α + 1) (sin α 1) = cos 2 α b). 1 + csc 2 (x) cos 2 (x) = csc 2 (x). c). cos x cos 3x cos x+cos 3x = tan 2x tan x 2

7 (4) Match each expressions a b with the correct expression in I V a). (1 + sin x) 2 + cos 2 x b). tan x cos x + csc x sin 2 x I). 2 II). 2 cos x III) sin x IV) cos x V). 2 sin xx (5) Without using calculator, find the exact value of the following: a). tan(15 o ) b) cos(165 ) c). sin(105 o ) 3

8 (6) Given that sin α = 2/5, α in quadrant IV; tan β = 4/3, β in quadrant III, find exact value of a). sin(α + β) b). tan(2β) (7) Given sin (α) = 1/3, and π/2 < α < π. a). State the quadrant where α 2 is located. b). Evaluate i). sin ( ) α 2 ii). cos ( ) α 2 iii). tan ( ) α 2 4

9 (8) a). Write the following in the form y = R sin(x + α). y = 2 sin x 2 cos x b). State the amplitude, phase shift and period of y 5

10 (9) Given f(x) = sec(2x π/2) + 2. a. Sketch one cycle of the functions f(x) b. State i). Domain: ii). Range: iii). Amplitude: iv). Period: v). Phase shift: vi). Vertical Asymptote 6

11 MTH 122 Plane Trigonometry Spring 2017 Test 3 Name INSTRUCTION: Answer all questions. Show your work and justify every steps of your work for full point. Circle your final answer. (1) (10pts) Find exact value of the following. Circle your final answer. a). csc 1 (2) b). cot 1 ( 1) c). sec 1 ( 2 ) d). sin ( cos 1 5 6). e). sin 1 ( sin 7π 6 ) 1

12 (2)(10pts) Find the general solution(s) of the following equations. Circle your final answer(s). a). 6 sin(x) + 3 = 4 b). cos(x) = 0 c).2 sin(x) 3 = 0 2

13 (3)(10pts) Find all real numbers that satisfy the following equations within the interval 0 x 2π. Circle your final answer(s). a). 2 cos(2x) 2 = 0. b). 3 tan (x π/3) =

14 (4) (15pts) Find all real numbers in the interval [0, 2π] satisfying the following equation. Circle your final answer a) 3 sin 2 (x) + 2 sin(x) 1 = 0 b) 6 sin 2 (x) 2 cos(x) = 5 4

15 (5) (10pts) Given a triangle(s) with the parts A = 28.6 o, b = 40.7, a = a). Solve the triangle completely. Clearly write out all parts derived b). Find the area of the triangle. Circle your final answer 5

16 (6)(15pts) Consider the triangle(s) with parts A = 41.2, b = 10.6cm, a = 8.1cm. a). Solve the triangle completely. Clearly write out all parts derived b). How many triangles are formed with the given parts? List parts of all triangles formed. 6

17 (7) (10pts) Solve the triangle with the given parts a = 10.3, c = 8.4, B = 88. 7

18 (8) (10pts) a). Find the area of a triangular piece of glass with sides 13in, 8in, and 9in. b). Find the area of the triangle with parts b = 8cm, c = 6cm, and A = 39. c). Find the area of the figure given below. 8

19 (9) (10pts) a). Draw the vector v = 7, 5. b). Find the magnitude and direction of vector v. 9

20 Marshall University MTH Plane Trigonometry Spring 2017 Final Exam Name Write your solutions in a clear and precise manner. 1. (20 pts) Given z = 3 4i and w = 2 + 3i. Find a). 4z 2w b). z w c). z 2 d). z w e). z and w f). Let θ and α be the argument of z and w, respectively. Find θ and α. 2. (15 pts) a). Convert z = i to trigonometric form b). Write z = 2 (cos(143 ) + i sin(143 )) in the form a + ib. Convert to 2d.p. 3. (15 pts) Given z = 4 (cos i sin 120 ) and w = 2 (cos 30 + i sin 30 ), evaluate the following in the form a + ib. a). z w b). z w 4. (15 pts) By first converting z = 2 + 2i to trigonometric form, use De Moivre s theorem to evaluate z 5. Leave your final answer in the form a + bi. 5. (15 pts) Find all of the fourth root of the complex number z = 4 4i. 6. (20 pts) For the vectors z = 4, 5 and w = 3, 2, perform the operations indicated. a). 3 w + 4 z c). magnitudes z and w. b). the dot product w z d). direction angle for z and w e). Find the smallest positive angle between the pair f). Sketch the two vectors on the same graph. g). Are the two vectors parallel, perpendicular or neither? 7. (10 pts) Find the vertical and horizontal component of the vector of magnitude 40 with direction angle (15 pts) Solve the triangles with parts a). B = 39, c = 11cm, b = 8cm. b). a = 10.3, c = 8.4, B = (10 pts) Find the area of the triangle with parts a = 22cm, b = 13cm, c = 18cm. 10. (15 pts) Find all real numbers in the interval [0, 360 ] that satisfy the equations a). 3 cos (3x) + 2 = 0. b). 3 sin 2 (x) + 2 sin(x) 1 = 0. 1