Questions: If cos(x) = 2/7, and x is in quadrant I, determine the exact value of each of the following. Use fractions and radicals only, no decimals. sin(x/2) = cos(x/2) = tan(x/2) =

Solution Steps

To solve this problem, we will use trigonometric identities and properties. Given that \(\cos(x) = \frac{2}{7}\) and \(x\) is in the first quadrant, we can find \(\sin(x)\) using the Pythagorean identity. Then, we will use the half-angle identities to find \(\sin\left(\frac{x}{2}\right)\), \(\cos\left(\frac{x}{2}\right)\), and \(\tan\left(\frac{x}{2}\right)\).

1. Find \(\sin(x)\): Use the identity \(\sin^2(x) + \cos^2(x) = 1\) to find \(\sin(x)\).
2. Half-Angle Identities:
   • \(\sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{2}}\)
   • \(\cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \cos(x)}{2}}\)
   • \(\tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)}\)

Using the Pythagorean identity, we find \(\sin(x)\) given \(\cos(x) = \frac{2}{7}\): \[ \sin^2(x) + \cos^2(x) = 1 \implies \sin^2(x) = 1 - \left(\frac{2}{7}\right)^2 = 1 - \frac{4}{49} = \frac{49}{49} - \frac{4}{49} = \frac{45}{49} \] Thus, \[ \sin(x) = \sqrt{\frac{45}{49}} = \frac{3\sqrt{5}}{7} \]

Using the half-angle identity for sine: \[ \sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{2}} = \sqrt{\frac{1 - \frac{2}{7}}{2}} = \sqrt{\frac{\frac{5}{7}}{2}} = \sqrt{\frac{5}{14}} = \frac{\sqrt{70}}{14} \]

Using the half-angle identity for cosine: \[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \cos(x)}{2}} = \sqrt{\frac{1 + \frac{2}{7}}{2}} = \sqrt{\frac{\frac{9}{7}}{2}} = \sqrt{\frac{9}{14}} = \frac{3\sqrt{14}}{14} \]

Using the definition of tangent: \[ \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} = \frac{\frac{\sqrt{70}}{14}}{\frac{3\sqrt{14}}{14}} = \frac{\sqrt{70}}{3\sqrt{14}} = \frac{\sqrt{5}}{3} \]

Final Answer