Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
Algebra 2/Trigonometry Regents, January 2013
Part I: Each correct answer will receive 2 credits.
11. If sin A = 1/3 , what is the value of cos 2A?
1) -2/3
2) 2/3
3) -7/9
4) 7/9
Answer: 4) 7/9
If you put cos(2 * sin-1(1/3)) into your calculator, you will get .7777..., which is 7/9.
Once of the double angle formulas for cosine is cos 2A = 1 - 2 sin2 A.
So cos 2A = 1 - 2(1/3)2 = 1 - 2(1/9) = 1 - 2/9 = 7/9
12. In the interval 0° < x < 360°, tan x is undefined when x equals
1) 0° and 90°
2) 90° and 180°
3) 180° and 270°
4) 90° and 270°
Answer: 4) 90° and 270°
Tan x = sin x / cos x, so Tan x is undefined when cos x = 0, which is when x is 90° and 270°.
On the Cartesian plane, tangent is undefined at the y-axis, which is the top and bottom of the unit circle.
13. If f(x) = SQRT(9 - x2), what are its domain and range?
1) domain: {x |-3 < x < 3}; range: {y|0 < y < 3}
2) domain: {x | x =/= +3}; range: {y|0 < y < 3}
3) domain: {x |-3 x < -3}; range: {y| y =/= 0}
4) domain: {x | x =/= +3}; range: {y|y > 0}
Answer: 1) domain: {x |-3 < x < 3}; range: {y|0 < y < 3}
The quadratic function g(x) = 9 - x2 opens downward from (0, 9) with zeroes at (-3,0) and (3,0).
However, any values of g(x) < 0 would cause f(g(x)) to be undefined. So the only valid values of x are -3 <. When x = 0, f(x) will be 3, so the range of the function will be between 0 and 3, inclusive. This is Choice (1).
Choice (2) says x cannot be + 3, which is not true because those are valid values. If the values of greater magnitude that aren't allowed.
Choice (3) contains the invalid numbers in its domain and excludes all the valid ones, except 3 and -3.
Choice (4) exclues 3, which is valid, but nothing else.
This could be solved by graphing the function in your calculator and then comparing the results to the notation in the choices.
14. When x2 + 3x - 4 is subtracted from x3 + 3x2 - 2x, the difference is
1) x3 + 2x2 - 5x + 4
2) x3 + 2x2 + x - 4
3) -x3 + 4x2 + x - 4
4) -x3 - 2x2 + 5x + 4
Answer: 1) x3 + 2x2 - 5x + 4
When subtracting the "FROM" clause goes on top. If I take $20 FROM my wallet, the amount of money that had been in my wallet would go on top.
Likewise, like subtracting a 3-digit number from a 4-digit number, you have to line the columns up correctly.
0x3 + x2 + 3x - 4
x3 + 2x2 - 5x + 4
Subtract each pair of terms, keeping subtraction of signed numbers in mind.
15. In the diagram below, the length of which line segment is equal to the exact value of sin θ?
1) TO
2) TS
3) OR
4) OS
Answer: 2) TS
In the Unit Circle, each point (x, y) on the circle has the coordinates (cos θ, sin θ), which makes sin θ the y-coordinate. The line segment ST runs from the x-axis (y = 0) to the point T on the circle (y = sin θ), and has a length of sin θ - 0 = sin θ.
The length of TO is 1, as it is the radius of the unit circle.
The length of OR is 1, as it goes from the origin to (1, 0).
The length of OS is cos θ, the x-coordinate of point T.
More to come. Comments and questions welcome.
More Regents problems.
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