Question

Consider this equation. tan ⁡ ( θ ) = - 5 3 if θ is an angle in quadrant ii, what is the value of sin ⁡ ( θ ) ? a. 3 ⁢ 34 34 b. - 3 ⁢ 34 34 c. 5 ⁢ 34 34 d.

Answer 1

The sine of an angle \( \theta \) in quadrant II, given tan(\( \theta \)) = -\frac{5}{3}, is found using the Pythagorean theorem to be -\frac{5\sqrt{34}}{34}. Since sine is positive in quadrant II, the correct answer is option (c) \( \frac{5}{\sqrt{34}} \).

Step-by-step explanation:

If we consider the equation tan(\( \theta \)) = -\frac{5}{3} and \( \theta \) is an angle in quadrant II, we need to find the value of sin(\( \theta \)). Recall that in quadrant II, sine is positive and tangent is negative. Since tangent is opposite over adjacent, we have an opposite side of -5 (negative indicates direction in quadrant II) and an adjacent side of 3 (also indicating direction in quadrant II).

We can use the Pythagorean theorem to find the hypotenuse: hypotenuse2 = opposite2 + adjacent2 which gives us hypotenuse2 = (-5)2 + (3)2, so the hypotenuse is \( \sqrt{34} \).

Since sine is opposite over hypotenuse, we get sin(\( \theta \)) = \frac{opposite}{hypotenuse} = \frac{-5}{\sqrt{34}}. We need to rationalize the denominator to get sin(\( \theta \)) = -\frac{5\sqrt{34}}{34}. Therefore, the correct answer is option (c), which is \( \frac{5}{\sqrt{34}} \).