Question

How do you evaluate \[\sec 120\] degrees?

Answer 1

Hint: Here, we will find the Trigonometric Value of the given Trigonometric Ratio. We will rewrite the given angle as the sum of two angles. Then we will use the trigonometric ratio to simplify the equation. Then we will use the trigonometric co-ratios and substitute the value of the trigonometric function of the obtained angle to get the required answer.

Formula Used:
We will use the following formula:
1. Trigonometric Ratio of \[90^\circ + \theta \] : \[\sec \left( {90^\circ + \theta } \right) = - \csc \theta \]
2. Trigonometric Co-ratio of \[\csc \theta = \dfrac{1}{{\cos \theta }}\]
3. Trigonometric Value of \[\cos 30^\circ = \dfrac{1}{2}\]

Complete step by step solution:
We are given a Trigonometric Ratio of \[\sec 120\] degrees.
\[ \sec 120^\circ = \sec \left( {90^\circ + 30^\circ } \right)\]
We know that Trigonometric Ratio of \[90^\circ + \theta \] :\[\sec \left( {90^\circ + \theta } \right) = - \csc \theta \] .
\[ \Rightarrow \sec 120^\circ = - \csc 30^\circ \]
We know that Trigonometric Co-ratio of \[\csc \theta = \dfrac{1}{{\cos \theta }}\].
Using this in the above equation, we get

\[ \Rightarrow \sec 120^\circ = - \dfrac{1}{{\cos 30^\circ }}\]
We know that the Trigonometric Value of \[\cos 30^\circ = \dfrac{1}{2}\]. Using this in above equation, we get
\[ \Rightarrow \sec 120^\circ = - \dfrac{1}{{\dfrac{1}{2}}}\]
\[ \Rightarrow \sec 120^\circ = - 2\]

Therefore, the value of \[\sec 120\] degrees is \[ - 2\].

Additional Information:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle. Trigonometric Ratios are used to find the relationships between the sides of a right-angle triangle. We should remember that all the trigonometric ratios can be written in the form of a basic Trigonometric Ratio of sine and cosine.

Note:
We should remember the rules that all the Trigonometric Ratios are positive in the First Quadrant. Sine and Cosecant are positive in the second quadrant and the rest are negative. Tangent and Cotangent are positive in the third quadrant and the rest are negative. Cosine and Secant are positive in the fourth quadrant and the rest are negative. This can be remembered as the ASTC rule in Trigonometry. This rule is used in determining the signs of the trigonometric ratio.