Question

How do you find the exact values of \[\arccos \left( {\dfrac{{\sqrt 3 }}{2}} \right)\]?

Answer 1

Hint:In the above question, is based on the inverse trigonometry concept. The trigonometric functions are the relationship between the angles and the sides of the triangle. Since measure is given in the function, we need to find the angle of that particular measure of trigonometric function.

Complete step by step solution:
arccos is an inverse trigonometric function which can also be written as \[{\cos ^{ - 1}}\]. -1 here is just the way of showing that it is inverse of \[\cos x\]. Inverse cosine does the opposite of cosine. Cosine function gives the angle which is calculated by dividing the adjacent side and hypotenuse in a right-angle triangle, but the inverse of it gives the measure of an angle.

The cosine of angle \[\theta \] is:
\[\cos \theta = \dfrac{{Adjacent}}{{Hypotenuse}}\]

Therefore, inverse of cosine is:
\[{\cos ^{ - 1}}\left( {\dfrac{{adjacent}}{{hypotenuse}}} \right) = \theta \]

In the above cosine function, we have to find inverse of cosine function with given value
\[\dfrac{{\sqrt
3 }}{2}\]
\[\arccos \left( {\dfrac{{\sqrt 3 }}{2}} \right) = \dfrac{\pi }{6}\]
\[\dfrac{\pi }{6}\] means 180 divided by 6 is 30 degrees. Since the right-angle triangle is a 30-60-
90 triangle So according to formula of cosine the hypotenuse is 2 then the adjacent side is \[\sqrt 3 \]

and the opposite side is 1 according to Pythagoras theorem. So, the angle whose cosine is \[\dfrac{{\sqrt 3
}}{2}\] is 30 degrees.

Note: An important thing to note is that since the value is a positive value the cosine function is in the first quadrant so it is easy to predict then angle will be between 0 to 90 degree. If you need to find out the value in fourth quadrant, we can subtract \[\dfrac{\pi }{6}\] by \[2\pi \] since the period of cosine is
\[2\pi \].