Question

Find \[\tan {240^ \circ }\],\[\cos {210^ \circ }\], \[\cos ( - {60^ \circ })\].

Answer 1

Hint: Learn how to calculate any angle. Learn the value of any trigonometric ratios in a certain quadrant. Learn how the value of the ratios changes with change in the quadrant to solve the problem. Then use the trigonometric table to get the value of each trigonometric function.

Complete step by step answer:
  We have given here to find the value of the three trigonometric ratios for different angles.
Now for the first ratio we have to find the value of the \[\tan {240^ \circ }\]. Now, if we rewrite the angle in multiple of right angles we can write it as,
\[{240^ \circ } = {90^ \circ } \times 3 + {60^ \circ }\].
So, we can see that the angle lies in the fourth quadrant and the angle is an odd multiple of \[{90^ \circ }\]. Hence, the value of \[\tan {240^ \circ }\]will be equal to \[\cot {60^ \circ }\].

Now, the value of tan in the fourth quadrant is negative since only cosine is positive in the fourth quadrant hence the sign will be negative. Hence, the value of \[\tan {240^ \circ }\] is,
\[\tan {240^ \circ } = - \cot {60^ \circ }\]
\[\Rightarrow \tan {240^ \circ } = - \dfrac{1}{{\sqrt 3 }}\]
Now, let's find the value of the \[\cos {210^ \circ }\]. Now, if we rewrite the angle in multiple of right angles we can write it as,
\[{210^ \circ } = {90^ \circ } \times 3 + {30^ \circ }\]
So, we can see that the angle lies in the fourth quadrant and the angle is an odd multiple of \[{90^ \circ }\]. Hence, the value of \[\cos {210^ \circ }\]will be equal to \[\sin {30^ \circ }\].

Now, the value cosine is positive in the fourth quadrant hence the sign will be positive.Hence, the value of \[\cos {210^ \circ }\]is,
\[\cos {210^ \circ } = \sin {30^ \circ }\]
\[\Rightarrow \cos {210^ \circ } = \dfrac{1}{2}\]
Now, we have to find the value of the \[\cos ( - {60^ \circ })\]. So, we can see that the angle lies in the fourth quadrant. Now, the value cosine is positive in the fourth quadrant hence the sign will be positive. Hence, the value of \[\cos ( - {60^ \circ })\] is,
\[\cos ( - {60^ \circ }) = \cos {60^ \circ }\]
\[\Rightarrow \cos {60^ \circ } = \dfrac{1}{2}\]

Hence the value of \[\tan {240^ \circ } = - \dfrac{1}{{\sqrt 3 }}\], \[\cos {210^ \circ } = \dfrac{1}{2}\], \[\cos {60^ \circ } = \dfrac{1}{2}\].

Note:To solve this type of problem remember the sign of the functions in different quadrants. Without the proper sign of the result the result will be incorrect. In the first quadrant value of all the functions are positive, in the second quadrant sine, cosine is positive, in third quadrant tan, cot is positive and in fourth quadrant cos, sec is only positive.