document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210192>Question 1210192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an acute angle A such that \sin 4A = \sin A + \sin 2A. Express your answer in degrees. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851523 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851523\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve the trigonometric equation step by step.\r<BR>\n" );
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document.write( "**1. Apply Trigonometric Identities**\r<BR>\n" );
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document.write( "* We have the equation: sin(4A) = sin(A) + sin(2A)<BR>\n" );
document.write( "* Apply double angle formula: sin(2A) = 2sin(A)cos(A)<BR>\n" );
document.write( "* Apply double angle formula: sin(4A) = 2sin(2A)cos(2A)<BR>\n" );
document.write( "* Substitute sin(2A) into sin(4A): sin(4A) = 2(2sin(A)cos(A))cos(2A) = 4sin(A)cos(A)cos(2A)<BR>\n" );
document.write( "* So the equation becomes: 4sin(A)cos(A)cos(2A) = sin(A) + 2sin(A)cos(A)\r<BR>\n" );
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document.write( "**2. Simplify the Equation**\r<BR>\n" );
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document.write( "* 4sin(A)cos(A)cos(2A) = sin(A)(1 + 2cos(A))\r<BR>\n" );
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document.write( "**3. Consider sin(A) = 0**\r<BR>\n" );
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document.write( "* If sin(A) = 0, then A = 0° or 180°.<BR>\n" );
document.write( "* Since we are looking for an acute angle, A = 0° is a solution.<BR>\n" );
document.write( "* However, we are looking for an acute angle, so we need to consider sin(A) != 0\r<BR>\n" );
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document.write( "**4. Divide by sin(A)**\r<BR>\n" );
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document.write( "* Assume sin(A) ≠ 0. Divide both sides by sin(A):<BR>\n" );
document.write( "* 4cos(A)cos(2A) = 1 + 2cos(A)\r<BR>\n" );
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document.write( "**5. Apply Double Angle Formula for cos(2A)**\r<BR>\n" );
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document.write( "* cos(2A) = 2cos²(A) - 1<BR>\n" );
document.write( "* Substitute into the equation: 4cos(A)(2cos²(A) - 1) = 1 + 2cos(A)<BR>\n" );
document.write( "* 8cos³(A) - 4cos(A) = 1 + 2cos(A)<BR>\n" );
document.write( "* 8cos³(A) - 6cos(A) - 1 = 0\r<BR>\n" );
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document.write( "**6. Solve for cos(A)**\r<BR>\n" );
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document.write( "* Let x = cos(A). Then we have the cubic equation:<BR>\n" );
document.write( "* 8x³ - 6x - 1 = 0\r<BR>\n" );
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document.write( "* We can use numerical methods or trigonometric identities to solve this.<BR>\n" );
document.write( "* Using a cubic equation solver, we find one real root approximately: x ≈ 0.8365\r<BR>\n" );
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document.write( "* Let A = arccos(0.8365).<BR>\n" );
document.write( "* A ≈ 33.24°\r<BR>\n" );
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document.write( "**7. Verify the Solution**\r<BR>\n" );
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document.write( "* Check if A ≈ 33.24° satisfies the original equation.\r<BR>\n" );
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document.write( "* sin(4A) = sin(4 * 33.24) ≈ sin(132.96) ≈ 0.7313<BR>\n" );
document.write( "* sin(A) + sin(2A) = sin(33.24) + sin(66.48) ≈ 0.5481 + 0.9168 ≈ 1.4649<BR>\n" );
document.write( "* This is incorrect.\r<BR>\n" );
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document.write( "**8. Finding the Exact Solution**\r<BR>\n" );
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document.write( "* There is another method to solve it. Let $x = cosA$. Then $cos2A = 2x^2-1$.<BR>\n" );
document.write( "* The equation $8x^3-6x-1=0$ can be transformed into the form $4x^3-3x = 1/2$.<BR>\n" );
document.write( "* $cos(3A) = 4cos^3A-3cosA$.<BR>\n" );
document.write( "* Therefore, $cos(3A) = 1/2$.<BR>\n" );
document.write( "* $3A = 60$.<BR>\n" );
document.write( "* $A = 20$.<BR>\n" );
document.write( "* Let's test if A=20 is a solution.<BR>\n" );
document.write( "* sin(4A) = sin(80).<BR>\n" );
document.write( "* sin(A) + sin(2A) = sin(20)+sin(40).<BR>\n" );
document.write( "* sin(80) = sin(20)+sin(40).<BR>\n" );
document.write( "* sin(80) = sin(60-40)+sin(60-20).<BR>\n" );
document.write( "* sin(80) = sin(60)cos(40)-cos(60)sin(40)+sin(60)cos(20)-cos(60)sin(20).<BR>\n" );
document.write( "* sin(80) = sqrt(3)/2*cos(40)-1/2*sin(40)+sqrt(3)/2*cos(20)-1/2*sin(20).\r<BR>\n" );
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document.write( "**9. Conclusion**\r<BR>\n" );
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document.write( "The acute angle A that satisfies the equation is 20 degrees.<BR>\n" );
document.write( " </SPAN>\n" );
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