How do you find the values of the six trigonometric functions of #t= ((7 pi) / 6)#?
Find values of trig functions
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To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for ( t = \frac{7\pi}{6} ), we can use the unit circle or the reference angle method.
Given that ( t = \frac{7\pi}{6} ), which lies in the third quadrant, we can determine its reference angle by subtracting ( \pi ) from ( \frac{7\pi}{6} ):
[ \text{Reference angle} = \frac{7\pi}{6}  \pi = \frac{\pi}{6} ]
Now, we can find the values of the trigonometric functions:

Sine (( \sin )): [ \sin\left(\frac{7\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} ]

Cosine (( \cos )): [ \cos\left(\frac{7\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} ]

Tangent (( \tan )): [ \tan\left(\frac{7\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} ]

Cosecant (( \csc )): [ \csc\left(\frac{7\pi}{6}\right) = \csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 ]

Secant (( \sec )): [ \sec\left(\frac{7\pi}{6}\right) = \sec\left(\frac{\pi}{6}\right) = \frac{1}{\cos\left(\frac{\pi}{6}\right)} = \frac{2}{\sqrt{3}} ]

Cotangent (( \cot )): [ \cot\left(\frac{7\pi}{6}\right) = \cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \sqrt{3} ]
So, the values of the six trigonometric functions for ( t = \frac{7\pi}{6} ) are:
[ \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}} ] [ \cot\left(\frac{7\pi}{6}\right) = \sqrt{3} ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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