Solve the equation for exact solutions over the interval calculator

#x_1=0#, #x_2=pi/3#, #x_3=pi#, #x_4=(5pi)/3# and thus the last option.

Depending on the model you use, there can be a variety of approaches to find zeros on a particular interval. If you are using a GDC like the TI-84, you might be able to determine zeros of the equation by defining, plotting, and analyzing the graph of the function #f(x)=sin 2x+sin x# (which equals to the left-hand side of the equation).
http://www.dummies.com/education/graphing-calculators/how-to-find-the-zeroes-of-a-function-with-the-ti-84-plus/

On the other hand, you could have been able to solve this equation by applying the doubling angle identity for the sine function,
#sin 2x=2sin x* cos x#

Therefore
#2sin x* cos x- sin x=0#

Factor out #sin x#
#sin x(2cos x-1)=0#

By the factor theorem the function would have a zero as long as at least one of these equation holds:
#sin x=0#
#cos x=1/2#.

Referring to a unit circle, along with #arcsin# and #arccos# functions on your calculator if necessary, and we find
#x_1=0#, #x_2=pi#, and
#x_3=pi/3#, #x_4=(5pi)/3#.

Evaluate these expressions on your calculator and ask for the decimal output to find the answer choice to this question. (Use #pi=3.14# if you are calculating by hand.)

You can verify these results by substituting the equation with the respective values of #x#. Alternatively, you can trace the graph to see if you get an #x# -intercept at these points.

Alright what you plug into your calculator will be inverse trig...
See below

Sin double angle identity:
#Sin2x=2SinxCosx#

#2SinxCosx-sinx=0#
Factor with GCF:
#sinx(2cosx-1)=0#
#sinx=0#
#2cosx-1=0#
#cosx=1/2#

You won't need inverse trig as these values are on the unit circle-
For #sinx=0#
#x=0, pi (3.14)#
For #cosx=1/2#
#x=pi/3 (1.05), (5pi)/3(5.24)#

The answer is the last option
0, 1.05, 3.14, 5.24

Because the domain given lists 0 as inclusive, the 0 stays as a solution

I've plugged into my calculator

solve
#(sin(2x)-sin(x)=0,x)| 0<=x<2pi#

#x in {0, pi/3, pi, 5*pi/3}#

Into decimals:
0, 1.05, 3.14, 5.24

sin 2x - sin x = 0
Using trig identity: sin 2x = 2sin x.cos x, we get:
2sin x.cos x - sin x = 0
sin x.(2cos x - 1) = 0
Either factor should be zero.
a. sin x = 0
Unit circle gives -->
x = 0, #x = pi#, and #x = 2pi# (rejected as outside of interval)
b. 2cos x - 1 = 0
#cos x = 1/2#
Trig table and unit circle give 2 solutions;
#x = pi/3#, and #x = (5pi)/3#
Answers for half closed interval [0, 2pi):
#0, pi/3; pi; (5pi)/3#
In radian:
[0, 1.05, 3.14, 5.24) -> Answer # 4

This calculator can solve basic trigonometric equations such as: $\color{blue}{ \sin(x) = \frac{1}{2} }$ or $ \color{blue}{ \sqrt{2} \cos\left(-\frac{3x}{4}\right) - 1 = 0 } $.

The calculator will find exact or approximate solutions on custom range. Solution can be expressed either in radians or degrees.

Exact trigonometric constants

Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. All values of sine, cosine, and tangent of angles with 3° increments are derivable using identities: Half-angle, Double-angle, Addition/subtraction and values for 0°, 30°, 36°, and 45°. Note that 1° = radians. This article is incomplete in at least two senses. First, it is always possible to apply a half-angle formula and find an e

1

Solved example of trigonometric equations

$8\sin\left(x\right)=2+\frac{4}{\csc\left(x\right)}$

2

The reciprocal sine function is cosecant: $\frac{1}{\csc(x)}=\sin(x)$

$8\sin\left(x\right)=2+4\sin\left(x\right)$

4

Eliminate the $4$ from the left side, multiplying both sides of the equation by the inverse of $4$

$\sin\left(x\right)=\frac{1}{2}$

5

The angles where the function $\sin\left(x\right)$ is $\frac{1}{2}$ are

$x=30^{\circ}+360^{\circ}n,\:x=150^{\circ}+360^{\circ}n$

6

The angles expressed in radians in the same order are equal to

$x=\frac{1}{6}\pi+2\pi n,\:x=\frac{5}{6}\pi+2\pi n$

Final Answer

$x=\frac{1}{6}\pi+2\pi n,\:x=\frac{5}{6}\pi+2\pi n$