- State the Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\).
- Substitute the given \(\sin \theta = \frac{16}{17}\) to find \(\cos^2 \theta = \frac{33}{289}\).
- Take the square root and identify the correct sign based on the quadrant: since \(\frac{\pi}{2} < \theta < \pi\), \(\cos \theta\) is negative.
- The final value is \(\cos \theta = \boxed{-\frac{\sqrt{33}}{17}}\).
Explanation
- Analysis of the Pythagorean Identity
We are given that \(\sin \theta = \frac{16}{17}\) and the angle \(\theta\) lies in the second quadrant, defined by the interval \(\frac{\pi}{2} < \theta < \pi\). We need to determine the value of \(\cos \theta\). The Pythagorean identity states that \(\sin^2 \theta + \cos^2 \theta = 1\), which will allow us to relate the sine and cosine of the same angle.
- Solving for \(\cos^2 \theta\)
To find \(\cos^2 \theta\), we substitute the known value \(\sin \theta = \frac{16}{17}\) into the identity:
\[\left(\frac{16}{17}\right)^2 + \cos^2 \theta = 1\]
\[\frac{256}{289} + \cos^2 \theta = 1\]
Subtracting \(\frac{256}{289}\) from both sides gives:
\[\cos^2 \theta = 1 - \frac{256}{289}\]
\[\cos^2 \theta = \frac{289 - 256}{289} = \frac{33}{289}\]
- Determining the Sign and Final Value
To find \(\cos \theta\), we take the square root of both sides:
\[\cos \theta = \pm \sqrt{\frac{33}{289}} = \pm \frac{\sqrt{33}}{17}\]
Since the angle \(\theta\) is in the second quadrant (\(\frac{\pi}{2} < \theta < \pi\)), the value of \(\cos \theta\) must be negative. Therefore, we discard the positive root and conclude that:
\[\cos \theta = -\frac{\sqrt{33}}{17}\]
Examples
The Pythagorean identity is fundamental in fields like engineering and physics. For instance, when analyzing the movement of a robotic arm or calculating the forces on a structural beam at a specific angle, you often need to resolve vectors into their components. If you know the vertical component (\(\sin \theta\)) and the total length or magnitude, using this identity allows you to calculate the horizontal component (\(\cos \theta\)) quickly, ensuring precision in design and mechanical movement.