How do you find the values of the six trigonometric functions of #t= ((7 pi) / 6)#?

Question

Oliver Atkinson · Accepted Answer

Find values of trig functions

Trig table and unit circle --> #sin t = sin ((7pi)/6) = sin (pi/6 + pi) = - sin (pi/6) = -1/2# #cos t = cos ((7pi)/6) = cos (pi/6 + pi) = - cos (pi/6) = -sqrt3/2# #tan t = sin/(cos) = (-1/2)(- 2/sqrt3) = 1/sqrt3 = sqrt3/3# #cot t = sqrt3/1 = sqrt3# #sec t = 1/(cos) = -2/sqrt3 = -2sqrt3/3# #csc t = 1/(sin) = -2#

Delilah Aguilar · Answer

undefined 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:

1. Sine ($ \sin $):
$$ \sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} $$

2. Cosine ($ \cos $):
$$ \cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$

3. Tangent ($ \tan $):
$$ \tan\left(\frac{7\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $$

4. Cosecant ($ \csc $):
$$ \csc\left(\frac{7\pi}{6}\right) = -\csc\left(\frac{\pi}{6}\right) = -\frac{1}{\sin\left(\frac{\pi}{6}\right)} = -\frac{1}{\frac{1}{2}} = -2 $$

5. Secant ($ \sec $):
$$ \sec\left(\frac{7\pi}{6}\right) = -\sec\left(\frac{\pi}{6}\right) = -\frac{1}{\cos\left(\frac{\pi}{6}\right)} = -\frac{2}{\sqrt{3}} $$

6. 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:

$$ \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} $$
$$ \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$
$$ \tan\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{3} $$
$$ \csc\left(\frac{7\pi}{6}\right) = -2 $$
$$ \sec\left(\frac{7\pi}{6}\right) = -\frac{2}{\sqrt{3}} $$
$$ \cot\left(\frac{7\pi}{6}\right) = -\sqrt{3} $$