Answer
Verified
417.9k+ views
Hint: We have to find the value of a negative angle of the secant function which is a trigonometric function. We shall use basic trigonometry to analyze the angle that has been referred to and then find its value in the secant function. For this, we must know the values of secant of the few principal angles in trigonometry derived by using the Pythagorean theory.
Complete step by step answer:
When we move in the anti-clockwise direction in the coordinate plane, the angles measured in this direction are all positive. However, when we move in the clockwise direction in the coordinate plane, the angles measured in this direction are considered to be all negative. Therefore, the negative angle given in the problem signifies that it has been measured in the clockwise sense.
We understand that $-\dfrac{\pi }{6}$ lies in the fourth quadrant of the cartesian plane which consists of angles ranging from $\left( \dfrac{3\pi }{2},2\pi \right)$ in the anticlockwise sense and angles ranging from $\left( 0,-\dfrac{\pi }{2} \right)$ in the clockwise sense.
According to the cartesian rule of sign of trigonometric functions, all the values of secant of angles lying in the fourth quadrant are positive.
This implies that $\sec \left( -\dfrac{\pi }{6} \right)$ must have a positive value. Thus, it can be written as $\sec \left( \dfrac{\pi }{6} \right)$.
Now, by the basic values of trigonometric functions, we know that $\sec \left( \dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$.
$\Rightarrow \sec \left( -\dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$
Therefore, the value of $\sec \left( -\dfrac{\pi }{6} \right)$ is $\dfrac{2}{\sqrt{3}}$.
Note: In the cartesian plane, the angles follow a particular sign system. According to this, all the trigonometric functions are positive in the first quadrant, only sine and cosecant functions are positive in the second quadrant, only tangent and cotangent functions are positive in the third quadrant and only cosine and secant functions are positive in the fourth quadrant.
Complete step by step answer:
When we move in the anti-clockwise direction in the coordinate plane, the angles measured in this direction are all positive. However, when we move in the clockwise direction in the coordinate plane, the angles measured in this direction are considered to be all negative. Therefore, the negative angle given in the problem signifies that it has been measured in the clockwise sense.
We understand that $-\dfrac{\pi }{6}$ lies in the fourth quadrant of the cartesian plane which consists of angles ranging from $\left( \dfrac{3\pi }{2},2\pi \right)$ in the anticlockwise sense and angles ranging from $\left( 0,-\dfrac{\pi }{2} \right)$ in the clockwise sense.
According to the cartesian rule of sign of trigonometric functions, all the values of secant of angles lying in the fourth quadrant are positive.
This implies that $\sec \left( -\dfrac{\pi }{6} \right)$ must have a positive value. Thus, it can be written as $\sec \left( \dfrac{\pi }{6} \right)$.
Now, by the basic values of trigonometric functions, we know that $\sec \left( \dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$.
$\Rightarrow \sec \left( -\dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$
Therefore, the value of $\sec \left( -\dfrac{\pi }{6} \right)$ is $\dfrac{2}{\sqrt{3}}$.
Note: In the cartesian plane, the angles follow a particular sign system. According to this, all the trigonometric functions are positive in the first quadrant, only sine and cosecant functions are positive in the second quadrant, only tangent and cotangent functions are positive in the third quadrant and only cosine and secant functions are positive in the fourth quadrant.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The only snake that builds a nest is a Krait b King class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Why is there a time difference of about 5 hours between class 10 social science CBSE
Which places in India experience sunrise first and class 9 social science CBSE