Find all solutions of the equation in 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).
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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

Solve 3sin x = 2cos^2 x

Ans: #pi/6 and (5pi)/6#

Explanation:

#3sin x = 2cos^2 x#
#3sin x = 2( 1 - sin^2 x) = 2 - 2sin^2 x#
#2sin^2 x + 3sin x - 2 = 0#
Solve this quadratic equation in sin x.
#D = d^2 = b^2 - 4ac = 9 + 16 = 25# --> #d = +- 5#
#sin x = -3/4 +- 5/4#
2 solutions --> sin x = 1/2 and sin x = - 2 (rejected as < -1).
#sin x = 1/2# --> #x = pi/6# and #x = (5pi)/6# (unit circle)

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Section 1-5 : Solving Trig Equations with Calculators, Part I

Find the solution(s) to the following equations. If an interval is given find only those solutions that are in the interval. If no interval is given find all solutions to the equation. These will require the use of a calculator so use at least 4 decimal places in your work.

1. \(7\cos \left( {4x} \right) + 11 = 10\) Solution
2. \(\displaystyle 6 + 5\cos \left( {\frac{x}{3}} \right) = 10\) in \(\left[ {0,38} \right]\) Solution
3. \(\displaystyle 3 = 6 - 11\sin \left( {\frac{t}{8}} \right)\) Solution
4. \(\displaystyle 4\sin \left( {6z} \right) + \frac{{13}}{{10}} = - \frac{3}{{10}}\) in \(\left[ {0,2} \right]\) Solution
5. \(\displaystyle 9\cos \left( {\frac{{4z}}{9}} \right) + 21\sin \left( {\frac{{4z}}{9}} \right) = 0\) in \(\left[ { - 10,10} \right]\) Solution
6. \(\displaystyle 3\tan \left( {\frac{w}{4}} \right) - 1 = 11 - 2\tan \left( {\frac{w}{4}} \right)\) in \(\left[ { - 50,0} \right]\) Solution
7. \(\displaystyle 17 - 3\sec \left( {\frac{z}{2}} \right) = 2\) in \(\left[ {20,45} \right]\) Solution
8. \(12\sin \left( {7y} \right) + 11 = 3 + 4\sin \left( {7y} \right)\) in \(\left[ { - 2, - \frac{1}{2}} \right]\) Solution
9. \(5 - 14\tan \left( {8x} \right) = 30\) in \(\left[ { - 1,1} \right]\) Solution
10. \(\displaystyle 0 = 18 + 2\csc \left( {\frac{t}{3}} \right)\) in \(\left[ {0,5} \right]\) Solution
11. \(\displaystyle \frac{1}{2}\cos \left( {\frac{x}{8}} \right) + \frac{1}{4} = \frac{2}{3}\) in \(\left[ {0,100} \right]\) Solution
12. \(\displaystyle \frac{4}{3} = 1 + 3\sec \left( {2t} \right)\) in \(\left[ { - 4,6} \right]\) Solution