Question

Find all 6 trigonometric ratios of the angle θ =7∏/ 6

Answer 1

check the picture below, those are the x,y pairs or cosine, sine values pair.


`\bf sin(\theta)=\cfrac{opposite}{hypotenuse} \qquad cos(\theta)=\cfrac{adjacent}{hypotenuse}\\\\ -------------------------------\\\\ cos\left( (7\pi )/(6) \right)=\cfrac{\stackrel{adjacent}{-√(3)}}{\stackrel{hypotenuse}{2}}\qquad \qquad sin\left( (7\pi )/(6) \right)=\cfrac{\stackrel{opposite}{1}}{\stackrel{hypotenuse}{2}}`

now, this is from the Unit Circle, and therefore the hypotenuse or radius wil be 1.


`\bf \begin{cases} adjacent=&-(√(3))/(2)\\ opposite=&(1)/(2)\\ hypotenuse=&1 \end{cases}\\\\ -------------------------------\\\\ sin(\theta)=\cfrac{opposite}{hypotenuse} \qquad cos(\theta)=\cfrac{adjacent}{hypotenuse} \quad % tangent tan(\theta)=\cfrac{opposite}{adjacent} \\\\\\ % cotangent cot(\theta)=\cfrac{adjacent}{opposite} \qquad % cosecant csc(\theta)=\cfrac{hypotenuse}{opposite} \quad % secant sec(\theta)=\cfrac{hypotenuse}{adjacent}`

so, just plug in those values... hmmm lemme do the tangent, so you see the division of two fractions.


`\bf tan\left( (7\pi )/(6) \right)=\cfrac{sin\left( (7\pi )/(6) \right)}{cos\left( (7\pi )/(6) \right)}\implies tan\left( (7\pi )/(6) \right)=\cfrac{(1)/(2)}{-(√(3))/(2)}\implies tan\left( (7\pi )/(6) \right)=\cfrac{1}{2}\cdot \cfrac{2}{-√(3)}`


`\bf tan\left( (7\pi )/(6) \right)=\cfrac{1}{-√(3)}\impliedby \textit{now, let's \underline{rationalize} the denominator} \\\\\\ \cfrac{1}{-√(3)}\cdot \cfrac{√(3)}{√(3)}\implies \cfrac{√(3)}{-(√(3))^2}\implies -\cfrac{√(3)}{3}`