Question

If \[\cos 3\theta = \dfrac{{\sqrt 3 }}{2}\] ; \[0 < \theta < 20^\circ \] , then the value of \[\theta \] is
A) \[15^\circ \]
B) \[10^\circ \]
C) \[0^\circ \]
D) \[12^\circ \]

Answer 1

Hint:
We will take the inverse of cosine on both sides of the equation. We will find the measure of angle \[3\theta \] and hence, the measure of angle \[\theta \] . We will convert the measure of the angle from radians to degrees using the formula for conversion.
Formulas used: We will use the following formulas:
1. \[{\cos ^{ - 1}}\left( {\cos x} \right)\] is equal to \[x\] when \[x\] belongs to the interval \[\left[ {0,\pi } \right]\] .
2. 180 degrees measure \[\pi \] radians: \[\pi {\rm{ radians}} = 180^\circ \].

Complete step by step solution:
We know that \[0 < \theta < 20^\circ \].
Not multiplying the above range by 3, we get
\[ \Rightarrow 0 < 3\theta < 60^\circ \]
The value of \[\cos 3\theta \] is \[\dfrac{{\sqrt 3 }}{2}\] .
Now taking \[{\cos ^{ - 1}}\] on both sides, we get
\[ \Rightarrow {\cos ^{ - 1}}\left( {\cos 3\theta } \right) = {\cos ^{ - 1}}\left( {\dfrac{{\sqrt 3 }}{2}} \right)\]
We know that \[{\cos ^{ - 1}}\left( {\cos x} \right)\] is equal to \[x\] when \[x\] belongs to the interval \[\left[ {0,\pi } \right]\] .
As \[0 < 3\theta < 60^\circ \] , we can say that:
\[\begin{array}{l} \Rightarrow {\cos ^{ - 1}}\left( {\cos 3\theta } \right) = {\cos ^{ - 1}}\left( {\dfrac{{\sqrt 3 }}{2}} \right)\\ \Rightarrow 3\theta = {\cos ^{ - 1}}\left( {\dfrac{{\sqrt 3 }}{2}} \right)\end{array}\]
Now, we know that \[{\cos ^{ - 1}}\dfrac{{\sqrt 3 }}{2}\] is equal to \[\dfrac{\pi }{6}\] . So,
 \[ \Rightarrow 3\theta = \dfrac{\pi }{6}\]
Dividing both sides by 3, we get
\[\begin{array}{l} \Rightarrow \dfrac{{3\theta }}{3} = \dfrac{\pi }{{6 \times 3}}\\ \Rightarrow {\rm{ }}\theta = \dfrac{\pi }{{18}}\end{array}\]
We have calculated that the measure of angle \[\theta \] is \[\dfrac{\pi }{{18}}\] radians. Now, we will convert this measure into degrees. We know that 180 degrees measure \[\pi \] radians:
\[ \Rightarrow \pi {\rm{ radians}} = 180^\circ \]
We will find the measure of 1 radian by dividing both sides of the above equation by \[\pi \]. Therefore, we get
 \[\begin{array}{l} \Rightarrow \dfrac{\pi }{\pi }{\rm{ radians}} = \dfrac{{180^\circ }}{\pi }\\ \Rightarrow {\rm{ }}1{\rm{ radian}} = \dfrac{{180^\circ }}{\pi }\end{array}\]
We will find the measure of \[\dfrac{\pi }{{18}}\] radians in degrees by multiplying both sides of the above equation by \[\dfrac{\pi }{{18}}\]. Therefore,
\[\begin{array}{l} \Rightarrow 1 \times \dfrac{\pi }{{18}}{\rm{radians}} = \dfrac{{180^\circ }}{\pi } \times \dfrac{\pi }{{18}}\\ \Rightarrow {\rm{ }}\dfrac{\pi }{{18}}{\rm{radians}} = 10^\circ = \theta \end{array}\]
 We have calculated the value of \[\theta \] to be \[10^\circ \] .

Hence option B is the correct option.

Note:
Here, we need to find the measure of the angle using trigonometric identity. So, it becomes important for us to remember basic trigonometric formulas and identities. We can also calculate the value of the angle if we remember the value of cosine of some standard angles. We know that the Cosine of 30 degrees is \[\dfrac{{\sqrt 3 }}{2}\] .
So, the measure of angle \[3\theta \] will also be 30 degrees and the measure of angle \[\theta \] will be one-third of 30 degrees; that is 10 degrees.