Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, June 2013)

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, June 2013

Part II: Each correct answer will receive 2 credits. Partial credit is available.


32. Find, algebraically, the measure of the obtuse angle, to the nearest degree, that satisfies the equation 5 csc θ = 8.

Answer:

Remember that csc = 1/sin.

5 csc θ = 8
5 = 8 sin θ
sin θ = 5/8
θ = sin^-1 (5/8) = 38.6...

Since 39 degrees would be an acute angle, we want 180 - 39 degrees = 141 degrees.

33. If g(x) = (ax * SQRT(1 - x))², express g(10) in simplest form.

Answer:

Subsitute and evaluate.

g(10) = (a(10) * SQRT(1 - (10)))²
= (10a * SQRT(-9))²
= (10a * 3i)²
= (30ai)²
= -900a

34. Express (cot x sin x) / sec x as a single trigonometric function, in simplest form, for all values of x for which it is defined.

Answer:

Replace cot with cos / sin and sec with 1 / cos.

(cot x sin x) / sec x
= (cos x / sin x ) * (sin x) * 1 / (1 / cos x)
= (cos x) * (cos x)
= cos² x

35. On a multiple-choice test, Abby randomly guesses on all seven questions. Each question has four choices. Find the probability, to the nearest thousandth, that Abby gets exactly three questions correct.

Answer:

There is a 1/4 chance that each question is correct, and a 3/4 chance that each question is wrong.

The probability that Q1,2,3 are correct and Q4,5,6,7 are incorrect is (1/4)(1/4)(1/4)(3/4)(3/4)(3/4)(3/4).

This number needs to be multiplied by the number of ways that exactly three correct answers can be chosen, which is ₇C₃.

So ₇C₃ (1/4)³ (3/4)⁴ = 0.173

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

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