Question 201357: I'm taking a practice test for the SAT II Subject Test in Math Level IC and there are all these questions about adding and subtracting sin, cos, tan, cos, sec, and cot which I have never learned. (I've taken Algebra I, Geometry, and Algebra II).
I know we're not supposed to ask about more than one problem in one question, but... here are the questions that I have no idea how to solve:
19. -cos^2 θ - 9sin^2 θ =
A. 0 B. 1 C. -1 D. 9 E. -9
34. If tan 10° = cot θ, then θ =
A. 10° B. 80° C. 70° D. -10° E. 90°
and
36. If cos (45+2x)° = sin (3x)°, then x=
A. 18° B. 27° C. 45° D. 22.5° E. 9°
Thanks for any help, even if you just help me with one of them!
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I'm sorry about my earlier answer. I clicked on "post your answer" accidentally and then was unable to find the question again so I could fix it.
a19. -cos^2 θ - 9sin^2 θ =
Factor out -1
-1*(cos^2 θ + 9sin^2 θ)
Separate 9sin^2 θ into sin^2 θ + 8sin^2 θ
-1*(cos^2 θ + sin^2 θ + 8sin^2 θ)
Replace cos^2 θ + sin^2 θ with a 1
-1*(1 + 8sin^2 θ)
Since sin^2 θ must be zero or positive, 8sin^2 θ will also be zero or positive. If you add 1 to zero you get 1, a positive number. If you add 1 to a positive number you get another positive number. So 1 + 8 sin^θ MUST be a positive number. And finally if you multiply a positive number by -1 you get a negative number.
So -1*(1 + 8sin^2 θ) must be a negative number. Answers C and E are the only negative answers. If θ = 0° then C is correct. If θ = 90° then the correct answer is E.
Did you leave out some restriction on the possible values of θ? Is there a reason to exclude 0° or 90°?
34. If tan 10° = cot θ, then θ =
Draw a right triangle and label one of the acute angles 10°. And label the "opposite" side for that angle as "x" and the "adjacent" side for that angle "y"
Since the tan function is the "opposite / adjacent" ratio, the tan 10° would be x/y.
What is the other acute angle in this triangle? (180° - 90° - 10° = 80°) What is the "opposite" side for the 80° angle? (y) What is the "adjacent" side for the 80° angle? (x). Since the cot function is the "adjacent / opposite" ratio, what is the cot of the 80° angle? (x/y).
Since the tan 10° and the cot 80° are both x/y they are equal to each other. The answer is B. In general, tan θ = cot (90° - θ)
36. If cos (45+2x)° = sin (3x)°, then x=
The key to this one is to use identities. The simplest way is probably to use cos y = sin (90° - y)
This says we can replace the cos of anything with the sin of (90° - anything). Therefore cos (45+2x)° = sin (90° - (45+2x)°) or sin (45 - 2x)°
Substituting into your problem we get
sin (45 - 2x)° = sin (3x)°
So the most obvious solution to this would be when
45 - 2x = 3x
Adding 2x to both sides we get
45 = 5x
Dividing by 5
9 = x
which is answer E.
Note: other possible solutions to sin (45 - 2x)° = sin (3x)° would be
45 - 2x = 3x + 360 ==> x = -63
45 - 2x = 3x + 720 ==> x = -135
45 - 2x = 3x - 360 ==> x = 81
etc.
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