Question

How do you find the value of cos 210degrees – csc 300 degrees?
(a) Using linear formulas
(b) Using trigonometric identities
(c) Using algebraic properties
(d) None of these

Answer 1

Hint: In this problem, we are to find the value of the given term cos 210degrees – csc 300 degrees. To start with, we will try to find out the value of $\cos 210{}^\circ $and $\csc 300{}^\circ $one after one and then get along with the solution. Getting both the values and simplifying them altogether will give us our needed result.

Complete step-by-step answer:
According to the question, we are to find the value of $\cos 210{}^\circ -\csc 300{}^\circ $.
So, we will try to deal with the term $\cos 210{}^\circ $first and then try simplifying it.
Now, 210 degrees can be written as 180 degrees + 30 degrees.
So, $\cos 210{}^\circ $can be written as, \[\cos 210{}^\circ =\cos \left( 180{}^\circ +30{}^\circ \right)\].
From the all sin tan cos formula of quadrants, we will get that the angle is in the third quadrant.
So, the value of the angle would be negative.
Then, we have now, \[\cos 210{}^\circ =\cos \left( 180{}^\circ +30{}^\circ \right)=-\cos 30{}^\circ \]
Again, from the trigonometric table, we get, \[\cos 30{}^\circ =\dfrac{\sqrt{3}}{2}\].
So, \[\cos 210{}^\circ =-\cos 30{}^\circ =-\dfrac{\sqrt{3}}{2}\].
Now, $\csc 300{}^\circ $can be written as, \[\cos 300{}^\circ =\cos \left( 270{}^\circ +30{}^\circ \right)\].
From the all sin tan cos formula of quadrants, we will get that the angle is in the fourth quadrant.
So, the value of the angle would be negative.
Then, we have now, \[\csc 300{}^\circ =\csc \left( 270{}^\circ +30{}^\circ \right)=-\sec 30{}^\circ \]
Again, from the trigonometric table, we get, \[\sec 30{}^\circ =\dfrac{2}{\sqrt{3}}\].
So, \[\csc 300{}^\circ =-\sec 30{}^\circ =-\dfrac{2}{\sqrt{3}}\].
Hence, we have now, $\cos 210{}^\circ -\csc 300{}^\circ =-\dfrac{\sqrt{3}}{2}-\left( -\dfrac{2}{\sqrt{3}} \right)=-\dfrac{\sqrt{3}}{2}+\dfrac{2}{\sqrt{3}}$
Now, simplifying, $\cos 210{}^\circ -\csc 300{}^\circ =\dfrac{-3+4}{2\sqrt{3}}=\dfrac{1}{2\sqrt{3}}$

So, the correct answer is “Option (b)”.

Note:To solve these kinds of problems, first we have to understand ASTC rules. The ASTC rule is nothing but the "all sin tan cos" rule in trigonometry. The angles which lie between 0° and 90° are said to lie in the first quadrant. The angles between 90° and 180° are in the second quadrant, angles between 180° and 270° are in the third quadrant and angles between 270° and 360° are in the fourth quadrant. In the first quadrant, the values for sin, cos and tan are positive. In the second quadrant, the values for sin are positive only. In the third quadrant, the values for tan are positive only. In the fourth quadrant, the values for cos are positive only.