Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x) < 0

Answer 1

The fact that csc(x) > 0 and tan(x) < 0 means that x is an angle in the 2nd quadrant.

Also, csc(x) = 8 means that sin(x) = 1/8.

Draw a right triangle in the 2nd quadrant whose vertical leg has length 1 and whose hypotenuse has length 8. Use the Pythagorean Theorem to find the length of the horizontal leg: -sqrt(8² - 1²) = -sqrt(63) (remember, we're in the 2nd quadrant, so we're on the negative horizontal axis).

So sin(x) = 1/8, cos(x) = -sqrt(63)/8, and tan(x) = -1/sqrt(63).

Use the Double Angle Formulas.

sin(2x) = 2sin(x)cos(x) = 2(1/8)(-sqrt(63)/8) = -sqrt(63)/32

cos(2x) = 1 - 2sin²(x) = 1 - 2(1/8)² = 31/32

tan(2x) = sin(2x)/cos(2x) = -sqrt(63)/31

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