Mathematics 5 SN TRIGONOMETRY PROBLEMS 1 If x 4 which one of the following statements is TRUE? A) sin x > 0 and cos x > 0 C) sin x < 0 and cos x > 0 B) sin x > 0 and cos x < 0 D) sin x < 0 and cos x < 0 If < t < which one of the following statements is TRUE? A) cos t > 0 and sin t > 0 C) cos t < 0 and sin t > 0 B) tan t > 0 and cos t < 0 D) sin t < 0 and tan t < 0 Given the functions f(x) = tan x and g(x) = sin x. What are the values for which the functions f and g are 0 in [0 ]? A) 0 C) 0 B) 0 D) 0
4 What is the solution set of sin x 1 = 0 in? A) {1} C) {x x = + n n Z} B) D) {x x = + n n Z}
What is the solution set of the equation sin x 4cos x + 1 = 0 where x -? A) 6 6 - C) 4 B) - D) 4 4 - If x [0 π[ find the solution set of the following equations : a) sin x + cos x = 1 b) sin x cos x = 0 Show your work. a) Work Result : The solution set is. 5 6
b) Work Result : The solution set is.
What are the zeros in the following equation if x [0 ]? sin x 1 = 0 A) 4-4 C) 4 7 4 B) 4 4 D) 4 7 4 5 4 4 Which values of x belonging to 0 satisfy the equation 4 sin x sin x = 0? The values of x belonging to 0 and that satisfy the equation are. 7 8
9 Sin x in the circle with centre O show on the right. P A Furthermore AB // OP and m OA = 1 unit. O x F E B What is the area of the shaded region? Show all your work. Show all your work. P A O x F E B Answer: The area of the shaded region is square units.
10 Find all the solutions of the equation cos x sin x = 0 such that x. Give your solution in radian measure. Show all your work. Show all your work. cos x sin x = 0 x. Answer: The values of x in the domain are radians.
11 What are the exact values of A that satisfy the following trigonometric equation? sin A cot A + cos A = 1 A [0 ] Show all your work. sin A cot A + cos A = 1 A [0 ] Answer: The solution is A =.
1 Given the following trigonometric equation: sin x + 5 sin x = 0 x [0 ] What are the exact values of x that satisfy this equation? The exact values of x are and.
1 Strange geometric formations known as crop circles have appeared in fields around the world. The creators of the crop circle shown below would like to surround their design with a border in the shape of an equilateral triangle. The circles that were used to make the design are congruent to the circle whose equation is: x + y 6x y 6 = 0 The circles are externally tangent to one another and tangent to the border. What is the length of the border? Round your answer to the nearest hundredth of a unit. Show all your work.
Show all your work. Answer: To the nearest hundredth of a unit the border measures.
14 Given the trigonometric equation: cos x + cos x = 1 x [ ] What are the exact values of x that satisfy this equation? The exact values of x satisfying the given equation are:.
15 A hyperbola and a trigonometric function are drawn on the same Cartesian plane. The equation of the hyperbola is x y 1 5 144. y F F 1 x The foci of the hyperbola are directly below two of the maxima of the trigonometric function. Which of the following is an equation of the trigonometric function? (Where necessary the numbers have been rounded.) A) y 8.8 cos x 1 1 C) y. cos x 11 11 B) y 8.8 cos x 1 1 D) y. cos x 11 11
16 Consider the equation cos sin = What are the solutions in radian measure to the equation for which 0? Show all your work. Show all your work. 0 cos sin = Answer In radian measure the exact answers are.
17 Given the following trigonometric equation: sin x + = 9 cos x x [0 ] What exact values of x satisfy this equation? The solution(s) to this equation is (are). 18 Given cos x + sin x = 0 x 0. What are the exact values of x? Show all your work. cos x + sin x = 0 x 0. Answer: The exact values of x are.
19 Prove the following trigonometric identity. sin x 1 cos x sin x 1 cos x Prove the following trigonometric identity. Show all your work. sin x 1 cos x sin x 1 cos x
0 Solve the following trigonometric equation. Give the exact value(s). sin x cos x = x 0 π Show all your work. sin x cos x = x 0 π Answer: The exact value(s) is (are).
1 Find all solutions of 6 sin x 5 sin x + 1 = 0 in the interval 0. Express any inexact solutions to the nearest hundredth. Show all your work. 6 sin x 5 sin x + 1 = 0 Solutions:
- Correction key 1 B D B 4 D 5 D
6 a) Work : (example) sin x + cos x = 1 1 cos x + cos x = 1 cos x cos x = 0 cos x (cos x 1) = 0 cos x = 0 or cos x = 1 x or or x = 0 Result : The solution set is 0. b) Work : (example) sin x cos x = 0 (1 cos x) cos x = 0 cos x cos x = 0 cos x + cos x = 0 cos x (cos x + 1) = 0-1 cos x = 0 or cos x =
x or or x or 4 Result : The solution set is 4 D The values of x belonging to 0 and that satisfy the equation are 6 5 6 0 and 6 7. 7 8
9 Example of an appropriate solution Measure of angle x P sin x = m x = sin 1 O x F A B E m x = 60 or m x = radians Base AB of triangle AOB m AB Height FO of triangle AOB m FO cos 60 1 Area of triangle AOB Area 1 0.4 square units 4 Area of sector AOB Area of sector AOB = (1) 10 1.05 square units 60 Area of the shaded region 1.05 0.4 = 0.6 square units
Answer: The area of the shaded region is 0.6 square units.
10 Example of an appropriate solution cos x sin x = 0 (1 sin x) sin x = 0 sin x sin x = 0 -sin x sin x 1 = 0 sin x + sin x + 1 = 0 (sin x + 1)(sin x + 1) = 0 sin x + 1 = 0 sin x + 1 = 0 sin x = -1 sin x = -1-1 sin x = x = 7 11 6 6 x = Answer The values of x in the domain are 11 and. 6
11 Example of an appropriate method sin A cot A + cos A = 1 sin A cos A sin A + cos A = 1 cos A + cos A = 1 cos A + cos A 1 = 0 ( cos A 1)(cos A + 1) = 0 cos A = 1 or cos A = -1 A = or 5 or A = Answer 5 1 5 The exact values of x are and. 6 6
1 Example of an appropriate method Equation of the circle in standard form x + 6x + y y = 6 (x ) + (y 1) = 6 + 9 + 1 (x ) + (y 1) = 6 Each radius measures 6 units Since the three circles are congruent the border forms an equilateral triangle. According to the diagram on the right A B C D Note: Adjustments will have to be made for different labelling. COD is a right triangle m OC = 6 units and m CDO = 0
m tan0 = m OC CD = m 6 CD m CD = 10.9 m AD = m AB + m BC + m CD = 10.9 + 1 + 10.9.78 Perimeter: P =.78 = 98.4 Answer: To the nearest hundredth of a unit the border measures 98.4. Note: Do not penalize students who did not round the answer. Students who determined the radius of the circle have shown a partial understanding of the problem. 14 The exact values of x are - - 15 A
16 Example of an appropriate solution cos θ sin θ 1 sin θ sin θ sin θ sin θ sin θ sin θ 1 0 sin θ 1sin θ 1 0 sin θ -1 or sin θ -1-1 For sin θ For sin θ -1 Reference angles: θ 6 θ The sine function is negative in Quadrants III and IV. Answer: 7 In radian measure the exact answers are θ 6 11 6 17 5 The exact values of x are and.
18 Example of an appropriate solution cos x sin 1 sin x sin sin x sin - sin sin x sin x sin x 0 x 0 x 0 x 1 0 x 1 0 sin x 1 sin x 1 0 sin x 1 0 sin x 1 = 0 sin x 1 sin x = 1 Answer: The exact values of x are x or 6 x.
19 Example of an appropriate proof sin x 1 sin x cos x 1 cos 1 cos 1 cos x1 cos x 1 cos x sin 1 cos x x 1 cos x x x 1 cos x 1 cos x 1 cos x 1 cos x Deduct marks if student is unable to complete the proof but is able to arrive at 1 cos 1 cos x x 0 Example of an appropriate solution sin 1 cos cos x cos x x cos x x cos x 0 cos 0 cos x 1cos x 1 1 cos x or x or x cos x 1 cos x 1 x π Answer: The exact values are x or x π.
Students who were able to arrive at 0 = ( cos x + 1) (cos x + 1) have shown they have a partial understanding of the problem. 1 Example of an appropriate solution 6 sin x 5 sin x 1 0 sin x 1 sin x 1 sin 0 sin x 1 0 1 x x 0.4 or.80 5 x 0.4.80 6 6 sin x 1 0 1 sin x x or 6 5 6 Note: Students who used an appropriate method to determine two of the four answers have shown they have a partial understanding of the problem.