Question

If \[\sin 2\theta = \cos 3\theta \] and \[\theta \] is an acute angle, then \[\sin \theta \]is equal to
A. \[\dfrac{{\sqrt 5 - 1}}{4}\]
B. \[\dfrac{{ - \sqrt 5 - 1}}{4}\]
C. \[0\]
D. none of these

Answer 1

Hint: In this problem, we are going to find the value \[\sin \theta \]by using trigonometry formulas. here is given as an acute angle and we should know the trigonometry angle table. In a right-angled triangle, the sin of an angle is equal to the ratio of the opposing side to the hypotenuse according to the sin theta formula.

Formula Used:The sine angle value is used. Hence \[\theta \]is an acute angle. the angle table is used according to the answer. From the trigonometry angle table\[\sin \dfrac{\pi }{{10}} = \dfrac{{\sqrt 5 - 1}}{4}\].

Complete step by step solution: From the given data
\[\sin 2\theta = \cos 3\theta \]
\[3\theta = 2n\pi \pm (\dfrac{\pi }{2} - 2\theta )\]
\[\theta = \dfrac{{2n\pi }}{5} + \dfrac{\pi }{{10}}\]
\[\theta = 2n\pi - \dfrac{\pi }{2}\] \[\theta \]is acute angle
\[\theta = \dfrac{\pi }{{10}}\]
\[\sin \theta = \dfrac{{\sqrt 5 - 1}}{4}\]
By substituting \[\theta \]value, \[\sin \theta = \dfrac{{\sqrt 5 - 1}}{4}\]

Option ‘A’ is correct

Note: Finding the values of trigonometric standard angles like \[0^\circ ,{\rm{ }}30^\circ ,{\rm{ }}45^\circ ,{\rm{ }}60^\circ ,\]and \[90^\circ \]is made easier by using the trigonometric ratios table. Trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent make up this system. As a result, it's important to keep in mind these common angles' trigonometric ratio values. Three sides make up a right-angled triangle: the hypotenuse, the opposite side (perpendicular), and the adjacent side (Base). The side with the greatest length is called the hypotenuse, the side perpendicular to the angle is called the opposite side, and the adjacent side is the side on which both the hypotenuse and the opposite side rest. It is crucial to identify the angle being considered while utilizing the trigonometric ratios. Most of the time, the answers to problems requiring the use of trigonometric tables to determine values result in constants. Keep this in mind when you double-check your response.