Question

How do you find the $6\,$ trigonometric functions of $330$ degrees?

Answer 1

Hint: Start by mentioning all the trigonometric relations and formula along with their respective signs. Reduce the terms on the both sides until they cannot be reduced any further if possible. Then finally evaluate the value of trigonometric functions.

Complete step-by-step solution:
First we will start off by applying the trigonometric relations.
$
  \sin (2\pi - \theta ) = - \theta \\
  \cos (2\pi - \theta ) = \theta \\
 $
Then next we will substitute the values in the identity.
$
 \sin (330) \\
    \Rightarrow \sin (360 - 30) \\
    \Rightarrow - \sin 30 \\
    \Rightarrow \dfrac{{ - 1}}{2} \\
 $
$
    \Rightarrow \cos (330) \\
    \Rightarrow \cos (360 - 30) \\
    \Rightarrow \cos 30 \\
    \Rightarrow \dfrac{{\sqrt 3 }}{2} \\
 $
Now we know the following trigonometric relations, hence we will start substituting values in these relations.
 \[
 \Rightarrow \tan x = \dfrac{{\sin x}}{{\cos x}} \\
 \Rightarrow \cot x = \dfrac{1}{{\tan x}} \\
 \Rightarrow \sec x = \dfrac{1}{{\cos x}} \\
 \Rightarrow \cos ecx = \dfrac{1}{{\sin x}} \\
 \]
Hence, the remaining values will be,
\[
 \Rightarrow \tan 330 = \dfrac{{\dfrac{{ - 1}}{2}}}{{\dfrac{{\sqrt 3 }}{2}}} = \dfrac{{ - 1}}{{\sqrt 3 }} \\
 \Rightarrow \cot x = \sqrt 3 \\
 \Rightarrow \sec x = \dfrac{2}{{\sqrt 3 }} \\
 \Rightarrow \cos ecx = - 2 \\
 \]

Hence, the $6$ values are $\dfrac{{ - 1}}{2},\dfrac{{\sqrt 3 }}{2},\dfrac{{ - 1}}{{\sqrt 3 }},\sqrt 3 ,\dfrac{2}{{\sqrt 3 }}, - 2$

Additional Information: To cross multiply terms, you will multiply the numerator in the first fraction times the denominator in the second fraction, then you write that number down. Then you multiply the numerator of the second fraction times the number in the denominator of your first fraction, and then you write that number down. By Cross multiplication of fractions, we get to know if two fractions are equal or which one is greater. This is especially useful when you are working with larger fractions that you are not sure how to reduce. Cross multiplication also helps us to solve for unknown variables in fractions.

Note: While substituting values of the terms, substitute the terms step-by-step to avoid any mistakes. Always take the variables to one side and integer type of terms to the other side. Also remember that$\sin {30^0}$ is $\dfrac{1}{2}$ and $\cos {30^0}$ is $\dfrac{{\sqrt 3 }}{2}$. Also, remember to mention all the values of the trigonometric functions in the end.