3. Given $$ heta=\frac{5 \pi}{6} $$, find all 6 trigonometric ratios.

Question

3. Given $$ 	heta=\frac{5 \pi}{6} $$, find all 6 trigonometric ratios.

UpStudy AI · Accepted Answer

1. The given angle is $$\theta=\frac{5\pi}{6}$$. Its reference angle is:
   $$
   \pi - \frac{5\pi}{6} = \frac{\pi}{6}
   $$

2. For $$\theta=\frac{5\pi}{6}$$ which lies in the second quadrant:
   - $$\sin \theta$$ is positive.
   - $$\cos \theta$$ is negative.
   - $$\tan \theta$$ is negative.

3. Find $$\sin \theta$$:
   $$
   \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi-\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
   $$

4. Find $$\cos \theta$$:
   $$
   \cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}
   $$

5. Find $$\tan \theta$$:
   $$
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
   $$

6. Find the reciprocal trigonometric ratios:
   - Secant:
     $$
     \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
     $$
   - Cosecant:
     $$
     \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2
     $$
   - Cotangent:
     $$
     \cot \theta = \frac{1}{\tan \theta} = -\sqrt{3}
     $$

7. Final answers:
   - $$\sin \theta = \frac{1}{2}$$
   - $$\cos \theta = -\frac{\sqrt{3}}{2}$$
   - $$\tan \theta = -\frac{\sqrt{3}}{3}$$
   - $$\sec \theta = -\frac{2\sqrt{3}}{3}$$
   - $$\csc \theta = 2$$
   - $$\cot \theta = -\sqrt{3}$$
The trigonometric ratios for $$\theta = \frac{5\pi}{6}$$ are: - $$\sin \theta = \frac{1}{2}$$ - $$\cos \theta = -\frac{\sqrt{3}}{2}$$ - $$\tan \theta = -\frac{\sqrt{3}}{3}$$ - $$\sec \theta = -\frac{2\sqrt{3}}{3}$$ - $$\csc \theta = 2$$ - $$\cot \theta = -\sqrt{3}$$

Given , find all 6 trigonometric ratios.

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