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How to find the exact value of \[\sec \left( { - \dfrac{\pi }{3}} \right)\] ?

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Last updated date: 19th Jul 2024
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Answer
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Hint: This problem is easy once we find the definition of sec. A secant of a curve is a line that intersects the curve at least two distinct points in geometry. Secant is derived from the Latin word secare, which means "to cut." A secant can intersect a circle at exactly two points in the case of a circle.

Complete step by step solution:
\[\sec = \dfrac{1}{{\cos }}\]
Let's start by converting to degrees from radians. The conversion for radians to degrees
\[\dfrac{{180}}{\pi }\].
\[
  \dfrac{{180}}{\pi } \times \left( { - \dfrac{\pi }{3}} \right) \\
   = \left( { - {{60}^\circ }} \right) \\
\]
To make this a positive angle, we must subtract 60 from 360, giving us \[{300^\circ }\] .
This is a unique perspective because it provides us with a precise response. However, we must first determine the reference angle before adding our special triangle. A reference angle is the angle between the terminal side of \[\theta \] to the x axis . It should always satisfy the interval \[{0^\circ } \leqslant \beta < {90^\circ }\] . The nearest x axis interception of \[{300^\circ }\] is at \[{360^\circ }\] . After subtracting, we arrive at a reference angle of \[{60^\circ }\] . We use the \[30 - 60 - 90\] , 1 , \[\sqrt 3 \] , 2 .
 Since 60 is greater than 30 , and \[{60^\circ }\] is the reference angle , this indicates that the side opposite our reference angle has been measured \[\sqrt 3 \] . The hypotenuse is always the longest, with a length of 2. As a result, we can deduce that the adjacent side is 1.
Applying the definition of cos:
adjacent/hypotenuse \[ = \] \[ - \dfrac{1}{2}\] (cos is negative in quadrant IV)
Substituting into sec .
1/(adjacent/hypotenuse) \[ = \] hypotenuse/adjacent \[ = - 2\]
Therefore, \[\sec \left( { - \dfrac{\pi }{3}} \right) = - 2\].

Note:
Sine, cosine, and tangent are the three most common trigonometric ratios. However, even though they are rarely used, there are three additional ratios: secant, cosecant, and cotangent. Since they can be conveniently calculated using the three key ratios, most calculators don't even have a button for them.