53 . D Draw the triangle to determine that side YZ is opposite X, and side XZ is opposite Y.
Using the law of sines, set up the proportion: . Multiply both sides by
sin57° to get YZ = . Choice (C) does not use the correct angle-side pairs, and (A),
(B), and (E) use the incorrect proportions.
56 . F The correct answer is (F). We can use SOHCAHTOA and a right triangle to defeat this
seemingly hard problem. Draw a right triangle and mark in angle θ. If cosθ = , that means
the adjacent side is 3, and the hypotenuse is 4. Now use the Pythagorean theorem to find the
third side: 3^2 + b^2 = 4^2 , so 9 + b^2 = 16, which simplifies to b^2 = 7 and finally to b = .
Your sketch should look something like this:
Now just plug your dimensions into the functions: sinθ tanθ = .
58 . J In the standard (x,y) coordinate plane, the sine of an angle in Quadrant II is always positive,
eliminating (F), (G), and (H). Make a right triangle with the x-axis, which gives you a 5-12-
13 triangle. The angle θ is at the origin, so the leg opposite the angle is 5 and the hypotenuse
is 13. Sine is defined as “opposite over hypotenuse,” which in this case gives , so the
answer is (J).
59 . D The range of sin x is normally between −1 to 1, inclusive; however, the range can change
depending on the graph’s amplitude and vertical shift. In the general form A sin(Bx + C) + D,
