IM STUCK ON THESE :(
1. What is the equation for shifting the standard sine curve +2 units horizontally?
A. y = sin (x + 2)
B. y = sin x + 2
C. y = sin x − 2
D. y = sin (x − 2)
3. What is tan-¹ √3/3 ?
A. π/4
B. -π/3
C. π/6
D. -π/4
4. cot–1 −0.57735 is approximately
A. −0.65.
B. −1.05.
C. 2.62.
D. 2.09.
5. What is the period of y=√3sin(1/3x-1/3)?
A. 6π
B. π/3
C. √3π
10. 16. Which of these statements are true??
(i) The domain of the inverse cosine function (y = cos–1 x) is –1 ≤ x ≤ 1.
(ii) The range of the inverse sine function (y = sin–1 x) is 0 ≤ y ≤ p.
(iii) The range of the inverse tangent function (y = tan–1 x) is –p/2 < y < p/2.
A. (i) and (ii)
B. (i), (ii), and (iii)
C. (ii) and (iii)
D. (i) and (iii)
11. What is the frequency of y=tan(π/3x)
A. 1/3
B. 3
C. 1/6
14. How many solutions does the system y=x
y=tan x have?
A. Infinitely many
B. 3
C. None
1. The equation for shifting the standard sine curve +2 units horizontally is y = sin (x - 2). So none of the given options are correct.
3. tan^(-1) √3/3 is equal to π/6. So the correct option is C.
4. cot^(-1) -0.57735 is approximately -0.65. So the correct option is A.
5. The period of y = √3sin(1/3x - 1/3) is 6π. So the correct option is A.
10. The correct statements are (i) and (iii). So the correct option is D.
11. The frequency of y = tan(π/3x) is 3. So the correct option is B.
14. The system y = x, y = tan x has infinitely many solutions. So the correct option is A.
1. To shift the standard sine curve +2 units horizontally, you need to subtract 2 from the x variable in the equation since this will shift the curve to the right. The correct equation is y = sin(x - 2). So, the answer is D.
3. To find tan-¹(√3/3), you need to find the angle whose tangent is √3/3. The correct answer is the angle whose tangent is √3/3. In the unit circle, this angle is π/6. So, the answer is C.
4. To find cot–1(-0.57735), you need to find the angle whose cotangent is -0.57735. The correct answer is the angle whose tangent is 1/-0.57735 = -1.73205. In the unit circle, this angle is approximately -1.05. So, the answer is B.
5. The period of the function y = √3sin(1/3x - 1/3) is given by 2π/|1/3|. Simplifying, the period is 6π. So, the answer is A.
10. (i) The domain of the inverse cosine function (y = cos–1 x) is -1 ≤ x ≤ 1. This is because the range of the cosine function is -1 ≤ x ≤ 1, and the inverse function switches the roles of the domain and range.
(ii) The range of the inverse sine function (y = sin–1 x) is -π/2 ≤ y ≤ π/2. This is because the range of the sine function is -1 ≤ x ≤ 1, and the inverse function switches the roles of the domain and range.
(iii) The range of the inverse tangent function (y = tan–1 x) is -π/2 < y < π/2. This is because the tangent function is periodic with period π, and the inverse function only takes values within one period.
So, the answer is D, (i) and (iii) are true.
11. The frequency of the function y = tan(π/3x) is determined by the coefficient of x inside the trigonometric function. In this case, the coefficient is 1/3. The frequency is the reciprocal of the coefficient, so the answer is 3. So, the answer is B.
14. The equation y = x represents a straight line, while y = tan(x) represents a periodic function. Since the straight line and the periodic function have different characteristics, they will have different solutions. Therefore, the system y = x and y = tan(x) does not have any solutions in common. So, the answer is C, None.