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What are the exact values of the six trigonometric functions for \( -\frac{7 \pi}{6} \) radians?

Ask by Thornton Gray. in the United States
Mar 11,2025

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The exact values of the six trigonometric functions for \( -\frac{7\pi}{6} \) radians are: \[ \begin{array}{rcl} \sin\left(-\frac{7\pi}{6}\right) & = & \frac{1}{2}, \\ \cos\left(-\frac{7\pi}{6}\right) & = & -\frac{\sqrt{3}}{2}, \\ \tan\left(-\frac{7\pi}{6}\right) & = & -\frac{\sqrt{3}}{3}, \\ \csc\left(-\frac{7\pi}{6}\right) & = & 2, \\ \sec\left(-\frac{7\pi}{6}\right) & = & -\frac{2\sqrt{3}}{3}, \\ \cot\left(-\frac{7\pi}{6}\right) & = & -\sqrt{3}. \end{array} \]

Solución

1. Compute a coterminal angle by adding \( 2\pi \): \[ -\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}. \] 2. The angle \( \frac{5\pi}{6} \) lies in the second quadrant where sine is positive and cosine is negative. Its reference angle is: \[ \pi - \frac{5\pi}{6} = \frac{\pi}{6}. \] 3. Using the known values for the reference angle \( \frac{\pi}{6} \): - \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \) - \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) - \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \) 4. Adjust these values for the second quadrant: - \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \) (sine is positive) - \( \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \) (cosine is negative) - \( \tan\left(\frac{5\pi}{6}\right) = \frac{\sin\left(\frac{5\pi}{6}\right)}{\cos\left(\frac{5\pi}{6}\right)} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \) 5. The reciprocal functions are: - \( \csc\left(-\frac{7\pi}{6}\right) = \csc\left(\frac{5\pi}{6}\right) = \frac{1}{\sin\left(\frac{5\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 \) - \( \sec\left(-\frac{7\pi}{6}\right) = \sec\left(\frac{5\pi}{6}\right) = \frac{1}{\cos\left(\frac{5\pi}{6}\right)} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \) - \( \cot\left(-\frac{7\pi}{6}\right) = \cot\left(\frac{5\pi}{6}\right) = \frac{1}{\tan\left(\frac{5\pi}{6}\right)} = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3} \) 6. In summary, the exact values of the six trigonometric functions for \( -\frac{7\pi}{6} \) radians are: \[ \begin{array}{rcl} \sin\left(-\frac{7\pi}{6}\right) & = & \frac{1}{2}, \\ \cos\left(-\frac{7\pi}{6}\right) & = & -\frac{\sqrt{3}}{2}, \\ \tan\left(-\frac{7\pi}{6}\right) & = & -\frac{\sqrt{3}}{3}, \\ \csc\left(-\frac{7\pi}{6}\right) & = & 2, \\ \sec\left(-\frac{7\pi}{6}\right) & = & -\frac{2\sqrt{3}}{3}, \\ \cot\left(-\frac{7\pi}{6}\right) & = & -\sqrt{3}. \end{array} \]

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To find the exact values of the six trigonometric functions for \( -\frac{7\pi}{6} \) radians, we can first recognize that this angle is located in the interior of the third quadrant when considering standard angle measures. The angle \( -\frac{7\pi}{6} \) is equivalent to \( \frac{5\pi}{6} \) when we add \( 2\pi \) (i.e., \( 2\pi - \frac{7\pi}{6} = \frac{5\pi}{6} \)). In the third quadrant, the reference angle is \( \frac{\pi}{6} \). Now, we can determine the values: 1. **Sine**: \[ \sin\left(-\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} \] 2. **Cosine**: \[ \cos\left(-\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \] 3. **Tangent**: \[ \tan\left(-\frac{7\pi}{6}\right) = \frac{\sin\left(-\frac{7\pi}{6}\right)}{\cos\left(-\frac{7\pi}{6}\right)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \] 4. **Cosecant**: \[ \csc\left(-\frac{7\pi}{6}\right) = -\csc\left(\frac{\pi}{6}\right) = -2 \] 5. **Secant**: \[ \sec\left(-\frac{7\pi}{6}\right) = -\sec\left(\frac{\pi}{6}\right) = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \] 6. **Cotangent**: \[ \cot\left(-\frac{7\pi}{6}\right) = \frac{\cos\left(-\frac{7\pi}{6}\right)}{\sin\left(-\frac{7\pi}{6}\right)} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} \] So, the exact values of the six trigonometric functions for \( -\frac{7\pi}{6} \) radians are: \[ \sin\left(-\frac{7\pi}{6}\right) = -\frac{1}{2}, \quad \cos\left(-\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}, \quad \tan\left(-\frac{7\pi}{6}\right) = \frac{\sqrt{3}}{3} \] \[ \csc\left(-\frac{7\pi}{6}\right) = -2, \quad \sec\left(-\frac{7\pi}{6}\right) = -\frac{2\sqrt{3}}{3}, \quad \cot\left(-\frac{7\pi}{6}\right) = \sqrt{3} \]

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