Questions: If cos(θ) = 1/4 and θ is in the 4th quadrant, find the exact value for sin(θ). sin(θ) =

If cos(θ) = 1/4 and θ is in the 4th quadrant, find the exact value for sin(θ). sin(θ) =
Transcript text: If $\cos (\theta)=\frac{1}{4}$ and $\theta$ is in the 4th quadrant, find the exact value for $\sin (\theta)$. \[ \sin (\theta)= \]
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Solution

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Solution Steps

Step 1: Identify the Quadrant and Use the Pythagorean Identity

Given that \(\cos(\theta) = \frac{1}{4}\) and \(\theta\) is in the 4th quadrant, we need to find \(\sin(\theta)\). In the 4th quadrant, the sine function is negative. We will use the Pythagorean identity:

\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]

Step 2: Substitute the Known Value

Substitute \(\cos(\theta) = \frac{1}{4}\) into the Pythagorean identity:

\[ \sin^2(\theta) + \left(\frac{1}{4}\right)^2 = 1 \]

\[ \sin^2(\theta) + \frac{1}{16} = 1 \]

Step 3: Solve for \(\sin^2(\theta)\)

Subtract \(\frac{1}{16}\) from both sides:

\[ \sin^2(\theta) = 1 - \frac{1}{16} \]

\[ \sin^2(\theta) = \frac{16}{16} - \frac{1}{16} = \frac{15}{16} \]

Step 4: Solve for \(\sin(\theta)\)

Take the square root of both sides. Since \(\theta\) is in the 4th quadrant, \(\sin(\theta)\) is negative:

\[ \sin(\theta) = -\sqrt{\frac{15}{16}} \]

\[ \sin(\theta) = -\frac{\sqrt{15}}{4} \]

Final Answer

\[ \boxed{\sin(\theta) = -\frac{\sqrt{15}}{4}} \]

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