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Trigonometric equations Calculator

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Example
Created on 2024-06-20Asked by Aria Williams (Solvelet student)
Solve the trigonometric equation cos⁡(2x)+3sin⁡(x)=0 \cos(2x) + \sqrt{3}\sin(x) = 0 for 0≤x≤2π 0 \leq x \leq 2\pi .

Solution

To solve the trigonometric equation cos⁡(2x)+3sin⁡(x)=0 \cos(2x) + \sqrt{3}\sin(x) = 0 : 1. \textbf{Express cos⁡(2x) \cos(2x) using a double-angle identity:} cos⁡(2x)=1−2sin⁡2(x) \cos(2x) = 1 - 2\sin^2(x) Substitute this into the original equation: 1−2sin⁡2(x)+3sin⁡(x)=0 1 - 2\sin^2(x) + \sqrt{3}\sin(x) = 0 2. \textbf{Let u=sin⁡(x) u = \sin(x) :} 1−2u2+3u=0 1 - 2u^2 + \sqrt{3}u = 0 3. \textbf{Rearrange the equation into standard quadratic form:} −2u2+3u+1=0 -2u^2 + \sqrt{3}u + 1 = 0 Multiply through by −1-1 to simplify: 2u2−3u−1=0 2u^2 - \sqrt{3}u - 1 = 0 4. \textbf{Solve the quadratic equation:} Using the quadratic formula u=−b±b2−4ac2a u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} , where a=2 a = 2 , b=−3 b = -\sqrt{3} , and c=−1 c = -1 : u=3±(3)2−4⋅2⋅(−1)2⋅2 u = \frac{\sqrt{3} \pm \sqrt{(\sqrt{3})^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} u=3±3+84 u = \frac{\sqrt{3} \pm \sqrt{3 + 8}}{4} u=3±114 u = \frac{\sqrt{3} \pm \sqrt{11}}{4} 5. \textbf{Find the possible values of u=sin⁡(x) u = \sin(x) :} u1=3+114 u_1 = \frac{\sqrt{3} + \sqrt{11}}{4} u2=3−114 u_2 = \frac{\sqrt{3} - \sqrt{11}}{4} Since sin⁡(x) \sin(x) must be between −1-1 and 1 1 , check the values: 3−114<−1(not valid) \frac{\sqrt{3} - \sqrt{11}}{4} < -1 \quad (\text{not valid}) Thus, the only valid solution is: sin⁡(x)=3+114 \sin(x) = \frac{\sqrt{3} + \sqrt{11}}{4} 6. \textbf{Solve for x x :} x=sin⁡−1(3+114) x = \sin^{-1}\left(\frac{\sqrt{3} + \sqrt{11}}{4}\right) Since sin⁡(x) \sin(x) must be in the range [0,2π] [0, 2\pi] , we have two solutions: x1=sin⁡−1(3+114) x_1 = \sin^{-1}\left(\frac{\sqrt{3} + \sqrt{11}}{4}\right) x2=π−sin⁡−1(3+114) x_2 = \pi - \sin^{-1}\left(\frac{\sqrt{3} + \sqrt{11}}{4}\right) 7. \textbf{Result:} The solutions to the equation cos⁡(2x)+3sin⁡(x)=0 \cos(2x) + \sqrt{3}\sin(x) = 0 for 0≤x≤2π 0 \leq x \leq 2\pi are: x=sin⁡−1(3+114)andx=π−sin⁡−1(3+114) x = \sin^{-1}\left(\frac{\sqrt{3} + \sqrt{11}}{4}\right) \quad \text{and} \quad x = \pi - \sin^{-1}\left(\frac{\sqrt{3} + \sqrt{11}}{4}\right) Solved on Solvelet with Basic AI Model
Some of the related questions asked by Daniel Martinez on Solvelet
1. Solve the equation sin⁡(x)=23\sin(x)=\frac{2}{3} for 0≤x≤2π0 \leq x \leq 2\pi.2. Determine the solutions to the equation 2cos⁡(2x)=3 2\cos(2x) = 3 for 0≤x≤2π 0 \leq x \leq 2\pi .
DefinitionA trigonometric equation consists of a trigonometric function, such as sine, cosine, or tangent, and is solved for the particular angle or value. Example: Solve sin(x)=21​. X=6π​+2kπ and x=65π​+2kπ
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