Question

If \[2{\cos ^2}x + 3\sin x - 3 = 0,0 \le x \le 180^\circ \], then \[x = \]
A. \[30^\circ ,90^\circ ,150^\circ \]
B. \[60^\circ ,120^\circ ,180^\circ \]
C. \[0^\circ ,30^\circ ,150^\circ \]
D. \[45^\circ ,90^\circ ,135^\circ \]

Answer 1

Hint
Trigonometry is the branch of mathematics that deals with particular angles' functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Trigonometry is used to establish directions like the north, south, east, and west. It also tells you the direction to point the compass in order to travel straight forward. To locate a certain location, it is used in navigation. It is also employed to calculate the separation between a location in the sea and the shore.
The link between a triangle's sides and angles is studied in trigonometry. The ratio of the side lengths is used to calculate the angle measurement using geometrical constructs. The sine function in trigonometry is the ratio of the hypotenuse's length to the opposite side's length in a right-angled triangle.
Formulaused:
Use trigonometry trivial identities
\[\begin{array}{l}{\sin ^2}x + {\cos ^2}x = 1\\{\cos ^2}x = 1-{\sin ^2}x\end{array}\]
Complete step-by-step solution
The given equation is ,
\[2{\cos ^2}x + 3\sin x - 3 = 0\]
With a ranged condition of \[0 \le x \le 180^\circ \]
So, the equation can be written as
\[2 - 2{\sin ^2}x + 3\sin x - 3 = 0\]
After solving this, the reduced equation is
\[ = > (2\sin x - 1)(\sin x - 1) = 0\]
\[ = > \sin x = \frac{1}{2}\]
Else, the equation can also be written as,
\[\sin x = 1\]
Then, the values of \[x\] is,
\[ x = > \frac{\pi }{2}\]
\[ = > \sin x = \frac{1}{2}\]
Then, the values of \[x\] is,
\[x = \frac{\pi }{6},\frac{{5\pi }}{6}\]
That is, it finally becomes
\[30^\circ ,90^\circ ,150^\circ \]
Hence, the correct option is A.

Note
Because you must learn several values of distinct functions in both degrees and radians, trigonometry is challenging. The sine function in trigonometry is the ratio of the hypotenuse's length to the opposite side's length in a right-angled triangle. To determine a right triangle's unknown angle or sides, utilize the sine function.
The cosine, denoted by the symbol \[cos\], is the proportion of the neighboring side of a right triangle to the hypotenuse.