This question tests understanding of the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ and how to use it to find missing trig values. The Pythagorean identity states that $\sin^2(\theta) + \cos^2(\theta) = 1$ for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form $\sin^2(\theta) = 1 - \cos^2(\theta)$ or $\cos^2(\theta) = 1 - \sin^2(\theta)$. Given $\tan(\theta) = \frac{4}{3}$, we can use a right triangle where opposite = 4, adjacent = 3, hypotenuse = 5, so $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}$ in magnitude; since θ is in Quadrant III, where cosine is negative, $\cos(\theta) = -\frac{3}{5}$. Choice B is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the negative sign. Choice A uses the wrong sign for cosine, forgetting that in Quadrant III, cosine is negative. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = 3/5, we use the identity to find cos²(θ) = 1 - sin²(θ) = 1 - (3/5)² = 1 - 9/25 = 16/25. Taking the square root gives cos(θ) = ±√(16/25) = ±4/5, and since θ is in Quadrant II, where cosine is negative, we choose cos(θ) = -4/5. Choice B is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the sign. Choice A uses the wrong sign for cosine, forgetting that in Quadrant II, cosine is negative. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity $ \sin^2(\theta) + \cos^2(\theta) = 1 $ and how to use it to find missing trig values. The Pythagorean identity states that $ \sin^2(\theta) + \cos^2(\theta) = 1 $ for any angle $ \theta $, which means that if you know one of these trig functions, you can find the other using the rearranged form $ \sin^2(\theta) = 1 - \cos^2(\theta) $ or $ \cos^2(\theta) = 1 - \sin^2(\theta) $. Given $ \cos(\theta) = \frac{4}{5} $, we rearrange the identity to $ \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left( \frac{4}{5} \right)^2 = 1 - \frac{16}{25} = \frac{9}{25} $, so $ \sin(\theta) = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} $. The quadrant information tells us sine is positive in Quadrant I, giving $ \sin(\theta) = \frac{3}{5} $. Choice A is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the positive sign. Choice C provides both $ \pm $ solutions when the quadrant information specifies a unique sign. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given cos(θ) = 8/17, we rearrange the identity to sin²(θ) = 1 - cos²(θ) = 1 - (8/17)² = 1 - 64/289 = 225/289, so sin(θ) = ±√(225/289) = ±15/17. The quadrant information tells us sine is negative in Quadrant IV, giving sin(θ) = -15/17. Choice C is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the negative sign. Choice B uses the wrong sign for sine, forgetting that in Quadrant IV, sine is negative. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = 1/2, we use the identity to find cos²(θ) = 1 - sin²(θ) = 1 - (1/2)² = 1 - 1/4 = 3/4. Taking the square root gives cos(θ) = ±√(3/4) = ±√3/2, and since θ is in Quadrant I, where cosine is positive, we choose cos(θ) = √3/2. Choice A is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the positive sign. Choice B uses the wrong sign for cosine, forgetting that in Quadrant I, cosine is positive. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = 3/5, we use the identity to find cos²(θ) = 1 - sin²(θ) = 1 - (3/5)² = 1 - 9/25 = 16/25. Choice B is correct because it properly applies the identity with correct arithmetic. Choice A makes an arithmetic error, calculating 9/25 instead of 16/25. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. The Pythagorean identity sin²(θ) + cos²(θ) = 1 is one of the most fundamental trig identities: it works for any angle, derives directly from either the unit circle or the Pythagorean theorem, and is essential for solving countless trig problems.

This question tests understanding of the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ and how to use it to find missing trig values. The Pythagorean identity states that $\sin^2(\theta) + \cos^2(\theta) = 1$ for any angle $\theta$, which means that if you know one of these trig functions, you can find the other using the rearranged form $\sin^2(\theta) = 1 - \cos^2(\theta)$ or $\cos^2(\theta) = 1 - \sin^2(\theta)$. Given $\sin(\theta) = \frac{5}{13}$, we use the identity to find $\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169}$. Taking the square root gives $\cos(\theta) = \pm \sqrt{\frac{144}{169}} = \pm \frac{12}{13}$, and since $\theta$ is in Quadrant II, where cosine is negative, we choose $\cos(\theta) = -\frac{12}{13}$. Choice B is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the negative sign. Choice A uses the wrong sign for cosine, forgetting that in Quadrant II, cosine is negative. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = -5/13, we use the identity to find cos²(θ) = 1 - sin²(θ) = 1 - (-5/13)² = 1 - 25/169 = 144/169. Taking the square root gives cos(θ) = ±√(144/169) = ±12/13, and since θ is in Quadrant III, where cosine is negative, we choose cos(θ) = -12/13. Choice B is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the negative sign. Choice C forgets to take the square root after finding cos²(θ) = 144/169, giving a wrong value. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = 3/5, we use the identity to find cos²(θ) = 1 - sin²(θ) = 1 - (3/5)² = 1 - 9/25 = 16/25. Taking the square root gives cos(θ) = ±√(16/25) = ±4/5, and since θ is in Quadrant II, where cosine is negative, we choose cos(θ) = -4/5. Choice B is correct because it properly applies the identity with correct arithmetic and uses the right quadrant to determine the negative sign. Choice A uses the wrong sign for cosine, forgetting that in Quadrant II, cosine is negative. Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. Remember the quadrant sign rules: Quadrant I (both positive), Quadrant II (sin positive, cos negative), Quadrant III (both negative), Quadrant IV (sin negative, cos positive)—use these to choose the correct sign after taking the square root.

This question tests understanding of the Pythagorean identity sin²(θ) + cos²(θ) = 1 and how to use it to find missing trig values. The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ, which means that if you know one of these trig functions, you can find the other using the rearranged form sin²(θ) = 1 - cos²(θ) or cos²(θ) = 1 - sin²(θ). Given sin(θ) = 8/17, we rearrange the identity to cos²(θ) = 1 - sin²(θ) = 1 - (8/17)² = 1 - 64/289 = 225/289. Choice B is correct because it properly applies the identity with correct arithmetic. Choice A makes an arithmetic error, calculating sin²(θ) instead of 1 - sin²(θ). Key to using the Pythagorean identity: when given one trig value (sin or cos), use the identity to find the other by rearranging to isolate the unknown, then take the square root and determine the correct sign based on which quadrant the angle is in. The Pythagorean identity sin²(θ) + cos²(θ) = 1 is one of the most fundamental trig identities: it works for any angle, derives directly from either the unit circle or the Pythagorean theorem, and is essential for solving countless trig problems.