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To find the exact values of the six trigonometric functions for [tex]\(-\frac{7 \pi}{6}\)[/tex] radians, let's start by identifying in which quadrant this angle is located and using reference angles.
### Step 1: Identify the Location in the Unit Circle
The angle [tex]\(-\frac{7 \pi}{6}\)[/tex] radians can be understood by first considering the positive equivalent angle [tex]\(2 \pi - \frac{7 \pi}{6}\)[/tex], which simplifies to:
[tex]\[ 2 \pi - \frac{7 \pi}{6} = \frac{12 \pi}{6} - \frac{7 \pi}{6} = \frac{5 \pi}{6} \][/tex]
Since [tex]\( -\frac{7 \pi}{6}\)[/tex] radians is obtained by rotating [tex]\(\pi\)[/tex] radians plus an additional [tex]\(\frac{\pi}{6}\)[/tex] radians in the negative direction (clockwise), it places this angle in the second quadrant where sine is positive, and cosine and tangent are negative.
### Step 2: Determine the Reference Angle
The reference angle for [tex]\( -\frac{7 \pi}{6} \)[/tex] is derived by subtracting [tex]\(\pi\)[/tex]:
[tex]\[ -\frac{7 \pi}{6} + 2 \pi = -\frac{7 \pi}{6} + \frac{12 \pi}{6} = \frac{5 \pi}{6} \][/tex]
### Step 3: Compute the Trigonometric Functions
Now, we use the exact trigonometric value for [tex]\(\frac{\pi}{6}\)[/tex] to find:
1. Sine
[tex]\[ \sin\left(-\frac{7 \pi}{6}\right) = \sin\left(\frac{\pi}{6} + \pi\right) = - \sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} \][/tex]
2. Cosine
[tex]\[ \cos\left(-\frac{7 \pi}{6}\right) = \cos\left(\frac{\pi}{6} + \pi\right) = - \cos\left(\frac{\pi}{6}\right)\][/tex]
[tex]\[ \cos\left(-\frac{7 \pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
3. Tangent
[tex]\[ \tan\left(-\frac{7 \pi}{6}\right) = \tan\left(\frac{\pi}{6} + \pi\right) = \tan\left(\frac{\pi}{6}\right) \][/tex]
[tex]\[ \tan\left(-\frac{7 \pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \][/tex]
4. Cosecant (reciprocal of sine)
[tex]\[ \csc\left(-\frac{7 \pi}{6}\right) = \frac{1}{\sin\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{1}{2}} = -2 \][/tex]
5. Secant (reciprocal of cosine)
[tex]\[ \sec\left(-\frac{7 \pi}{6}\right) = \frac{1}{\cos\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \][/tex]
6. Cotangent (reciprocal of tangent)
[tex]\[ \cot\left(-\frac{7 \pi}{6}\right) = \frac{1}{\tan\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{\sqrt{3}}{3}} = -\frac{3}{\sqrt{3}} = -\sqrt{3} \][/tex]
### Conclusion
The exact values of the six trigonometric functions for [tex]\(-\frac{7 \pi}{6}\)[/tex] radians are:
- [tex]\(\sin\left(-\frac{7 \pi}{6}\right) = 0.5\)[/tex]
- [tex]\(\cos\left(-\frac{7 \pi}{6}\right) = -0.8660254037844388\)[/tex]
- [tex]\(\tan\left(-\frac{7 \pi}{6}\right) = -0.5773502691896254\)[/tex]
- [tex]\(\csc\left(-\frac{7 \pi}{6}\right) = 2.0000000000000013\)[/tex]
- [tex]\(\sec\left(-\frac{7 \pi}{6}\right) = -1.1547005383792512\)[/tex]
- [tex]\(\cot\left(-\frac{7 \pi}{6}\right) = -1.7320508075688783\)[/tex]
### Step 1: Identify the Location in the Unit Circle
The angle [tex]\(-\frac{7 \pi}{6}\)[/tex] radians can be understood by first considering the positive equivalent angle [tex]\(2 \pi - \frac{7 \pi}{6}\)[/tex], which simplifies to:
[tex]\[ 2 \pi - \frac{7 \pi}{6} = \frac{12 \pi}{6} - \frac{7 \pi}{6} = \frac{5 \pi}{6} \][/tex]
Since [tex]\( -\frac{7 \pi}{6}\)[/tex] radians is obtained by rotating [tex]\(\pi\)[/tex] radians plus an additional [tex]\(\frac{\pi}{6}\)[/tex] radians in the negative direction (clockwise), it places this angle in the second quadrant where sine is positive, and cosine and tangent are negative.
### Step 2: Determine the Reference Angle
The reference angle for [tex]\( -\frac{7 \pi}{6} \)[/tex] is derived by subtracting [tex]\(\pi\)[/tex]:
[tex]\[ -\frac{7 \pi}{6} + 2 \pi = -\frac{7 \pi}{6} + \frac{12 \pi}{6} = \frac{5 \pi}{6} \][/tex]
### Step 3: Compute the Trigonometric Functions
Now, we use the exact trigonometric value for [tex]\(\frac{\pi}{6}\)[/tex] to find:
1. Sine
[tex]\[ \sin\left(-\frac{7 \pi}{6}\right) = \sin\left(\frac{\pi}{6} + \pi\right) = - \sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} \][/tex]
2. Cosine
[tex]\[ \cos\left(-\frac{7 \pi}{6}\right) = \cos\left(\frac{\pi}{6} + \pi\right) = - \cos\left(\frac{\pi}{6}\right)\][/tex]
[tex]\[ \cos\left(-\frac{7 \pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
3. Tangent
[tex]\[ \tan\left(-\frac{7 \pi}{6}\right) = \tan\left(\frac{\pi}{6} + \pi\right) = \tan\left(\frac{\pi}{6}\right) \][/tex]
[tex]\[ \tan\left(-\frac{7 \pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \][/tex]
4. Cosecant (reciprocal of sine)
[tex]\[ \csc\left(-\frac{7 \pi}{6}\right) = \frac{1}{\sin\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{1}{2}} = -2 \][/tex]
5. Secant (reciprocal of cosine)
[tex]\[ \sec\left(-\frac{7 \pi}{6}\right) = \frac{1}{\cos\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \][/tex]
6. Cotangent (reciprocal of tangent)
[tex]\[ \cot\left(-\frac{7 \pi}{6}\right) = \frac{1}{\tan\left(-\frac{7 \pi}{6}\right)} = \frac{1}{- \frac{\sqrt{3}}{3}} = -\frac{3}{\sqrt{3}} = -\sqrt{3} \][/tex]
### Conclusion
The exact values of the six trigonometric functions for [tex]\(-\frac{7 \pi}{6}\)[/tex] radians are:
- [tex]\(\sin\left(-\frac{7 \pi}{6}\right) = 0.5\)[/tex]
- [tex]\(\cos\left(-\frac{7 \pi}{6}\right) = -0.8660254037844388\)[/tex]
- [tex]\(\tan\left(-\frac{7 \pi}{6}\right) = -0.5773502691896254\)[/tex]
- [tex]\(\csc\left(-\frac{7 \pi}{6}\right) = 2.0000000000000013\)[/tex]
- [tex]\(\sec\left(-\frac{7 \pi}{6}\right) = -1.1547005383792512\)[/tex]
- [tex]\(\cot\left(-\frac{7 \pi}{6}\right) = -1.7320508075688783\)[/tex]
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