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\n" ); document.write( "Solving Trig equations like this usually involves three stages:
- Use algebra and/or Trig properties to transform the equation into one or more equations of the form:
\n" ); document.write( "TrigFunction(expression) = number - Find the general solution for the equation(s) from stage 1. Trig. functions are periodic so there are usually an infinite number of solutions to these equations. The general solution expresses these infinite solutions.
- Often these problems ask for a specific solution, like \"Find the smallest positive solution to...\" or \"Find all solutions that are between p and q\". To find specific solutions, you use the general solution.
\n" ); document.write( "1. Transform.
\n" ); document.write( "With this equation the transformation stage is simple. Just divide both sides by 5:
\n" ); document.write( "
\n" ); document.write( "2. Find the general solution.
\n" ); document.write( "We should recognize that 3/5 is not a special angle value for sin. So we will need our calculators. First we'll convert 3/5 to s decimal:
\n" ); document.write( "sin(2x) = 0.6
\n" ); document.write( "Then we will use our calculators to find the reference angle. Making sure your calculator is set to degree mode you should get:
\n" ); document.write( "
\n" ); document.write( "Since this sin value is positive, it tells that the angle, 2x, is in the 1st or 2nd quadrants (where sin's are positive.) Combining the reference angle with the quadrants we get the equations:
\n" ); document.write( "2x = 36.86989765 + 360n (for the 1st quadrant)
\n" ); document.write( "2x = 180 - 36.86989765 + 360n (for the second quadrant)
\n" ); document.write( "with the second equation simplifying to:
\n" ); document.write( "2x = 143.13010235 + 360n
\n" ); document.write( "Now we solve for x by dividing both equations by 2:
\n" ); document.write( "x = 18.43494882 + 180n
\n" ); document.write( "x = 71.56505118 + 180n
\n" ); document.write( "Rounding to 2 decimal places (as the problem instructs) we get:
\n" ); document.write( "x = 18.43 + 180n
\n" ); document.write( "x = 71.57 + 180n
\n" ); document.write( "This is the general solution.
\n" ); document.write( "(Note: The \"n\" in these equations is a placeholder for any integer. To get specific x values, you replace \"n\" with an integer. Each integer value for n will result in an x value that is a solution to our equation. Some books/teachers use a different letter, like \"k\". The specific letter used is not important. What is important is that no matter what letter is used, it is a placeholder for an integer.)
\n" ); document.write( "3. Find the specific solution(s).
\n" ); document.write( "This is where we actually replace the n's in the general solution equations. We keep trying different n's until we feel we have found all the specific solutions. \"[0 degrees, 360 degrees)\" means
\n" ); document.write( "x = 18.43 + 180n
\n" ); document.write( "When n = 0, x = 18.43
\n" ); document.write( "When n = 1, x = 198.43
\n" ); document.write( "When n = 2 or larger, x is more than 360 (too big for
\n" ); document.write( "When n is negative, x is negative (too small for
\n" ); document.write( "x = 71.57 + 180n
\n" ); document.write( "When n = 0, x = 71.57
\n" ); document.write( "When n = 1, x = 251.57
\n" ); document.write( "When n = 2 or larger, x is more than 360 (too big for
\n" ); document.write( "When n is negative, x is negative (too small for
\n" ); document.write( "So all our specific solutions, in order from lowest to highest, are:
\n" ); document.write( "18.43, 71.57, 198.43 and 251,57. \n" ); document.write( "