User: You are given access to make searches on the Internet. The format for making a search is: 1. The user sends a message. 2. You will decide whether to make a search. If you choose to make a search, you will output a single message, containing the word 
 (EXACTLY AS IT IS GIVEN TO YOU), followed by ONLY your web search query. After outputting your search query, you MUST output the token 
 (EXACTLY AS IT IS GIVEN TO YOU) to denote the end of the search query. Keep your search query concise as there is a 400 characters limit.
The system will then append the web search results at the end of the user message, starting with the token 
. Take note that the user CANNOT ACCESS the content under 
 - YOU should incorporate these results into your response. Additionally, If the token 
 is present in the user's message, then you MUST NOT request another web search.

IMPORTANT! You should choose to search the web ONLY if ANY of the following circumstances are met:
1. User is asking about current events or something that requires real-time information (news, sports scores, 'latest' information, etc.)
2. User is asking about some term you are unfamiliar with (e.g. if it's a little-known term, if it's a person you don't know well, etc.)
3. User is asking about anything that involves numerical facts (e.g. distances between places, population count, sizes, date and time, etc.)
4. User explicitly asks you to search or provide links to references.
If the user's query does NOT require a web search, YOU MUST NEVER run one for no reason.
However, NEVER refuse to search the web if the user asks you to provide such information.

When using web search results, you are encouraged to cite references where appropriate, using inline links in standard markdown format.
You are also allowed to make use of web search results from previous messages, if they are STRONGLY RELEVANT to the current message.

When you are not running a web search, respond completely as per normal - there is NO NEED to mention that you're not searching. 
Do your best to be informative. However, you MUST NOT make up any information. If you think you might not know something, search for it.

Assistant: Understood. I will strictly follow these instructions in this conversation.
User: Show that x = pi/10 is one of the roots of the equation sin3x = cos2x


 for "Show that x = pi/10 is one of the roots of the equation sin3x = cos2x":

1.
URL: https://math.stackexchange.com/questions/1617890/solve-sin3x-cos2x
Title: Solve $&#92;&#92;sin(3x)=&#92;&#92;cos(2x)$ - Mathematics Stack Exchange
Content: Use <b>sin</b> <b>3x</b> <b>=</b> 3 sin <b>x</b> − 4sin3 <b>x</b> and cos 2x <b>=</b> 1 − 2sin2 <b>x</b>. To get. 3 sin <b>x</b> − 4sin3 <b>x</b> <b>=</b> 1 − 2sin2 <b>x</b>. Now call sin <b>x</b> <b>=</b> t. Thus we have. 4t3 − 2t2 − 3t + 1 <b>=</b> 0. Observe that t <b>=</b> 1 is definitely a solution, so we have. (t − 1)(4t2 + 2t − 1) <b>=</b> 0. The quadratic factor will be zero for.

2.
URL: https://www.symbolab.com/solver/trigonometric-equation-calculator
Title: Trigonometric Equation Calculator - Symbolab
Content: Use inverse trigonometric functions to find the solutions, and check for extraneous solutions. A basic trigonometric <b>equation</b> has the form sin (x)=a, cos (x)=a, tan (x)=a, cot (x)=a. The formula to convert radians to degrees: degrees <b>=</b> radians * 180 / π.

3.
URL: https://www.mathway.com/trigonometry
Title: Trigonometry Problem Solver - Mathway
Content: Free math problem solver answers your trigonometry homework questions with step-by-step explanations.

4.
URL: https://socratic.org/trigonometry/trigonometric-identities-and-equations/solving-trigonometric-equations
Title: Solving Trigonometric Equations - Trigonometry - Socratic
Content: To solve a trig <b>equation</b>, transform it into <b>one</b>, or many, basic trig <b>equations</b>. Solving a trig <b>equation</b>, finally, results in solving various basic trig <b>equations</b>. There are 4 main basic trig <b>equations</b>: sin <b>x</b> <b>=</b> a; cos <b>x</b> <b>=</b> a; tan <b>x</b> <b>=</b> a; cot <b>x</b> <b>=</b> a. Exp. Solve sin 2x - 2sin <b>x</b> <b>=</b> 0 Solution. Transform the <b>equation</b> into 2 basic trig <b>equations</b>:

5.
URL: https://math.libretexts.org/Workbench/Algebra_and_Trigonometry_2e_(OpenStax)/09%3A_Trigonometric_Identities_and_Equations/9.06%3A_Solving_Trigonometric_Equations
Title: 9.6: Solving Trigonometric Equations - Mathematics LibreTexts
Content: Analysis. We can see the solutions on the graph in Figure 3.On the interval 0≤θ<2π,0≤θ<2π, the graph crosses the x-axis four times, at the solutions noted.Notice that trigonometric <b>equations</b> <b>that</b> are in quadratic form can yield up to four solutions instead of the expected two that are found with quadratic <b>equations</b>.

6.
URL: https://www.shaalaa.com/question-bank-solutions/find-general-solution-following-equation-sin-3-x-cos-2-x-0_58599
Title: Find the General Solution of the Following Equation: Sin 3 X + Cos 2 X ...
Content: If \[2 \sin^2 <b>x</b> <b>=</b> 3\cos <b>x</b>\]. where \[0 \leq <b>x</b> \leq 2\pi\], then find the value of <b>x</b>. If a is any real number, the number of <b>roots</b> <b>of</b> \[\cot <b>x</b> - \tan <b>x</b> <b>=</b> a\] in the first quadrant is (are). In (0, π), the number of solutions of the <b>equation</b> \[\tan <b>x</b> + \tan 2x + \tan 3x <b>=</b> \tan <b>x</b> \tan 2x \tan 3x\] is

7.
URL: https://socratic.org/trigonometry/trigonometric-identities-and-equations/proving-identities
Title: Proving Identities - Trigonometry - Socratic
Content: <b>The</b> Trigonometric Identities are <b>equations</b> <b>that</b> are true for Right Angled Triangles. Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2π while tangent and cotangent have period π. Identities for negative angles. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions.

8.
URL: https://www.symbolab.com/solver/step-by-step/sin3x%2Bcos2x%3D0
Title: sin3x+cos2x=0 - Symbolab
Content: Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step

9.
URL: https://mathsolver.microsoft.com/en/solve-problem/%60sin%20x%20%2B%20%60sin%203%20x%20%3D%201%20%2B%20%60cos%202%20x
Title: Solve sinx+sin3x=1+cos2x | Microsoft Math Solver
Content: Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.

10.
URL: https://math.stackexchange.com/questions/3242816/sin-x-sin2x-sin3x-cos-x-cos2x-cos3x
Title: $&#92;&#92;sin x + &#92;&#92;sin2x + &#92;&#92;sin3x = &#92;&#92;cos x + &#92;&#92;cos2x + &#92;&#92;cos3x$
Content: $\begingroup$ @DiyaSarkar You divided out $2\cos x+1$ at <b>one</b> point, but you can't do that if $2\cos x+1=0$, so you have to check those <b>roots</b> too. $\endgroup$ - J.G. Commented May 28, 2019 at 14:22

11.
URL: https://www.teachoo.com/2192/584/Ex-3.3--20---Prove-sin-x---sin-3x---sin2-x---cos2-x--2-sin-x/category/Ex-3.3/
Title: Ex 3.3, 20 - Prove sin x - sin 3x / sin2 x - cos2 x = 2 sin x - Teachoo
Content: Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo. Ex 3.3, 20 Prove that 〖sin <b>x</b> − 〗⁡<b>sin⁡3x</b> / (sin2⁡x − <b>cos2⁡x</b> ) <b>=</b> 2 sin <b>x</b> Solving L.H.S. 〖sin <b>x</b> −〗⁡<b>sin⁡3x</b> / (sin2⁡x − cos2 ...

12.
URL: https://socratic.org/questions/how-do-you-find-all-cos2x-sin3x-sinx-in-the-interval-0-2pi
Title: How do you find all (cos2x)/(sin3x-sinx) in the interval [0,2pi ...
Content: Graph reads y values, <b>x</b> in [ 0, 2pi ] sans indeterminate holes at <b>x</b> <b>=</b> pi/4, 3/4pi, 5/4pi and 7/4pi , and asymptotic <b>x</b> <b>=</b> 0, pi and 2pi y <b>=</b> (cos 2x)/(sin 3x - sin <b>x</b> ...

13.
URL: https://www.symbolab.com/solver/roots-calculator
Title: Roots Calculator - Symbolab
Content: A <b>root</b> <b>is</b> a value for which the function equals zero. The <b>roots</b> are the points where the function intercept with the x-axis. Complex <b>roots</b> are the imaginary <b>roots</b> <b>of</b> a function. To find the complex <b>roots</b> <b>of</b> a quadratic <b>equation</b> use the formula: <b>x</b> <b>=</b> (-b±i√ (4ac - b2))/2a. High School Math Solutions - Radical <b>Equation</b> Calculator.

14.
URL: https://www.desmos.com/calculator?lang=en-GB
Title: Graphing Calculator - Desmos
Content: Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic <b>equations</b>, add sliders, animate graphs, and more.

15.
URL: https://math.stackexchange.com/questions/3759476/solving-sin-x-cos3x-cos-x-sin3x-sqrt3-2-in-the-interval-0-2-pi
Title: algebra precalculus - Solving $&#92;sin x&#92;cos3x+&#92;cos x&#92;sin3x=&#92;sqrt {3}/2 ...
Content: Solve the following trig <b>equation</b>: $$\sin(x)\cos(3x)+\cos(x)\sin(3x)=\frac{\sqrt{3}}{2}$$ in the interval $[0,2\pi]$. Using trig identities, it is now at $$\sin(x+3x ...


Assistant: