Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


6. In △ABC, m∠A = 74, a = 59.2, and c = 60.3. What are the two possible values for m∠C, to the nearest tenth?

1) 73.7 and 106.3
2) 73.7 and 163.7
3) 78.3 and 101.7
4) 78.3 and 168.3

Answer: 3) 78.3 and 101.7

The Law of Sines says that Sin A / c = Sin C / c

So Sin C = c Sin A / a

Sin C = 60.3 Sin 74 / 59.2 = 0.9791...

C = 78.3, which is 90 - 11.7. The sine of (90 + 11.7) is also 0.9791.

So Choice (3) 78.3 and 101.7 are the correct answers.

7. What is the principal value of cos^-1(√(3)/2)?

1) −30°
2) 60°
3) 150°
4) 240°

Answer: 3) 150°

Where on the Unit Circle is the x value = -√(3)/2? At 150 degrees and 210 degrees.

The principal value is 150°, which is Choice (3).

8. What is the value of x in the equation 9^{3x + 1} = 27^{x + 2}?

1) 1
2) 1/3
3) 1/2
4) 4/3

Answer: 4) 4/3

If you aren't sure how to approach this, yo can always substitute and evaluate each of the choices using your calculator.

Both 9 and 27 are powers of 3, so each expression can be rewritten to have 3 as its base. If the two equal expressions have the same base then there exponents must be equal as well.

9^{3x + 1} = 27^{x + 2}

(3²)^{3x + 1} = (3³)^{x + 2}

3^{6x + 2} = 3^{3x + 6}

6x + 2 = 3x + 6

3x = 4

x = 4/3

The correct answer is Choice (4).

9. The roots of the equation 2x² + 7x - 3 = 0 are

1) -1/2 and -3
2) 1/2 and 3
3) (-7 + √(73))/4
4) (7 + √(73))/4

Answer: 3) (-7 + √(73))/4

A quick check tells you that 18 - 21 - 3 =/= 0 and 18 + 21 - 3 =/= 0. So Choices (1) and (2) are eliminated.

If you have to use the Quadratic Formula, the first thing is -b, which is -7. That is Choice (3) and not Choice (4).

10. Which ratio represents csc A in the diagram below?

1) 25/24
2) 25/7
3) 24/7
4) 7/24

Answer: 2) 25/7

The ratio csc is the inverse of sin. Since sin is OPP/HYP, then csc is HYP/OPP.

The hypotenuse is 25. The side opposites from angle A is 7. So csc A is 25/7, which is Choice (2).

Choice (1) is HYP/ADJ, which is 1/cos, which is sec.

Choice (3) is ADJ/OPP, which is 1/tan, which is cot.

Choice (4) is OPP/ADJ, which is tan.

More to come. Comments and questions welcome.

More Regents problems.

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