alleryStart learning →Given tan x = -3/4 with x in quadrant II, find sin(2x).
A24 / 25
B-24 / 25
C-7 / 25
D7 / 25
Answer & Solution
Correct answer: B. -24 / 25
1. In quadrant II the adjacent side is negative, so take opposite 3 and adjacent -4.
2. The Pythagorean Theorem gives (-4)^2 + 3^2 = 16 + 9 = 25, so the hypotenuse is 5.
3. That makes sin x = 3/5 and cos x = -4/5.
4. Apply sin(2x) = 2 sin x cos x.
5. Substituting gives 2 times (3/5) times (-4/5) = -24/25.
6. The positive 24/25 ignores the negative cosine that quadrant II forces.
7. The value 7/25 is cos(2x) for this angle, so it answers a different part of the problem.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_
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