Final answer:
Using trigonometric double angle formulas and the given value of tan(x) in the fourth quadrant, we can calculate sin(2x) as −24/25, cos(2x) as 7/25, and tan(2x) as −48/7 without solving for x directly.
Step-by-step explanation:
If tan(x) = −¾ and x is in quadrant IV, we know that the cosine of x will be positive and the sine of x will be negative because that is the nature of the trigonometric functions in the fourth quadrant. To find the exact values of the expressions without solving for x, we use the trigonometric identities for double angles.
Solution for (a) sin(2x):
Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x). Since we know tan(x) = sin(x)/cos(x), we can find sin(x) and cos(x) using the given information that tan(x) = −¾. We do this by considering the right triangle where the opposite side is −3 (negative because sine is negative in the fourth quadrant) and the adjacent side is 4 (positive because cosine is positive in the fourth quadrant). By the Pythagorean theorem, the hypotenuse will be −5. Therefore, sin(x) = −3/−5, and cos(x) = 4/−5. Plugging these into the double angle formula, we get sin(2x) = 2(−3/−5)(4/−5) = −24/25.
Solution for (b) cos(2x):
Using the double angle formula for cosine, cos(2x) = cos^2(x) - sin^2(x). Substituting the previously found sin(x) and cos(x), we get cos(2x) = (4/−5)^2 - (−3/−5)^2 = 16/25 - 9/25 = 7/25.
Solution for (c) tan(2x):
Using the double angle formula for tangent, tan(2x) = 2tan(x)/(1 - tan^2(x)). Substituting tan(x) = −3/4, we get tan(2x) = 2(−3/4)/(1 - (−3/4)^2) = −3/2/(1 - 9/16) = −3/2/(7/16) = −⅔/2×7/16 = −⅔/2×7/16 = −⅔8/7.