document.write( "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).
\n" ); document.write( "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:
\n" ); document.write( "19. -cos^2 θ - 9sin^2 θ =
\n" ); document.write( "A. 0 B. 1 C. -1 D. 9 E. -9\r
\n" ); document.write( "\n" ); document.write( "34. If tan 10° = cot θ, then θ =
\n" ); document.write( "A. 10° B. 80° C. 70° D. -10° E. 90°\r
\n" ); document.write( "\n" ); document.write( "and\r
\n" ); document.write( "\n" ); document.write( "36. If cos (45+2x)° = sin (3x)°, then x=
\n" ); document.write( "A. 18° B. 27° C. 45° D. 22.5° E. 9°\r
\n" ); document.write( "\n" ); document.write( "Thanks for any help, even if you just help me with one of them! \n" ); document.write( "
Algebra.Com's Answer #151577 by jsmallt9(3758)\"\" \"About 
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.
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\n" ); document.write( "\n" ); document.write( " a19. -cos^2 θ - 9sin^2 θ =
\n" ); document.write( "Factor out -1
\n" ); document.write( "-1*(cos^2 θ + 9sin^2 θ)
\n" ); document.write( "Separate 9sin^2 θ into sin^2 θ + 8sin^2 θ
\n" ); document.write( "-1*(cos^2 θ + sin^2 θ + 8sin^2 θ)
\n" ); document.write( "Replace cos^2 θ + sin^2 θ with a 1
\n" ); document.write( "-1*(1 + 8sin^2 θ)
\n" ); document.write( "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.
\n" ); document.write( "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.
\n" ); document.write( "Did you leave out some restriction on the possible values of θ? Is there a reason to exclude 0° or 90°?
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\n" ); document.write( "\n" ); document.write( "34. If tan 10° = cot θ, then θ =
\n" ); document.write( "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\"
\n" ); document.write( "Since the tan function is the \"opposite / adjacent\" ratio, the tan 10° would be x/y.
\n" ); document.write( "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).
\n" ); document.write( "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)
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\n" ); document.write( "\n" ); document.write( "36. If cos (45+2x)° = sin (3x)°, then x=
\n" ); document.write( "The key to this one is to use identities. The simplest way is probably to use cos y = sin (90° - y)
\n" ); document.write( "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)°
\n" ); document.write( "Substituting into your problem we get
\n" ); document.write( "sin (45 - 2x)° = sin (3x)°
\n" ); document.write( "So the most obvious solution to this would be when
\n" ); document.write( "45 - 2x = 3x
\n" ); document.write( "Adding 2x to both sides we get
\n" ); document.write( "45 = 5x
\n" ); document.write( "Dividing by 5
\n" ); document.write( "9 = x
\n" ); document.write( "which is answer E.
\n" ); document.write( "Note: other possible solutions to sin (45 - 2x)° = sin (3x)° would be
\n" ); document.write( "45 - 2x = 3x + 360 ==> x = -63
\n" ); document.write( "45 - 2x = 3x + 720 ==> x = -135
\n" ); document.write( "45 - 2x = 3x - 360 ==> x = 81
\n" ); document.write( "etc. \n" ); document.write( "
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