Question 607379
{{{(2/3) sec(x) = cos(x)}}}
Solving Trig equations like this usually involve 3 steps:<ol><li>Use algebra and/or Trig properties to transform the given equation into one or more equations of the form:
TrigFunction(expression) = number</li><li>Find the general solution(s) for the equation(s) from step 1. Given that all the Trig functions are periodic, these general solutions will express an infinite set of solutions.</li><li>If the problem asks for a specific set of solutions (e.g. "the smallest positive" or "solutions on the interval 0 to {{{2*pi}}}") then use the general solution equation(s) from step 2 to find them.</li></ol>Let's see this in action:
1. Transform the equation into the desired form. Since your equation has more than one Trig function you probably want to use Trig properties to try to get it down to one function. To so this we can either<ul><li>Replace sec(x) with 1/cos(x) (this would make the equation a "cos-only" equation); or</li><li>Multiply both sides of the equation by cos(x). Since cos and sec are reciprocals of each other and since the product of reciprocals is always a 1, the left side will simplify to just 2/3.</li></ul>
Both of these will work. I'm going to use the second option primarily because there will be no functions in a denominator to deal with:
{{{cos(x)*((2/3) sec(x)) = cos(x)*(cos(x))}}}
which simplifies to:
{{{2/3 = cos^2(x)}}}
Except for the exponent, we nearly have the desired form. All we have to do now is find the square root of each side (and don't forget the negative square root!):
{{{cos(x) = sqrt(2/3) = 0.81649658}}} or {{{cos(x) = -sqrt(2/3) = -0.81649658}}}
We now have two equations of the desired form.<br>
2. Find the general solution. This involves:<ol><li>Finding the reference angle for each equation from step 1</li><li>Determining the quadrants for the angles in each equations from step 1</li><li>Determining angles which would have the proper reference angle and which terminate in the proper quadrants.</li><li>Write equations of the general solution in the form:
expression = angle + {{{2*pi*n}}}
where "expression" is from TrigFunction(expression) = number and "angle" is an angle determined in the previous step. There will an equation in the general solution for each angle determined in the previous step.</li><li>If "expression" is not simply a variable (e.g. sin(2x) = ...), then solve the equations for the variable.</li></ol>Let's see this in action.
2.1) Find the reference angle(s). For {{{cos(x) = 0.81649658}}} we find {{{cos^(-1)(0.81649658)}}} on our calculator. (Assuming "x" usually means radian measure, be sure your calculator is set for radian measure.) You should get 0.61547971 for the reference angle. Since the sign is ignored when finding a reference angle, we end up with the same reference angle for cos(x) = -0.81649658.
2.2) Find the quadrants. For the equation where the cos is positive, the angle must terminate in the 1st or 4th quadrants. For the equation where the cos is negative, the angle must terminate in the 2nd or 3rd quadrants.
2.3) Find angles with the proper reference angles in the proper quadrants. For the first quadrant, you can use the reference angle: 0.61547971. For the 4th quadrant you can use the negative of the reference angle (or you can subtract the reference angle from {{{2pi}}}). For the second quadrant, subtract the reference angle from {{{pi}}}: {{{pi - 0.61547971 = 2.52611294}}}. For the third quadrant, add the reference angle to {{{pi}}}: {{{pi + 0.61547971 = 3.75707236}}}.
2.4) Write equations:
{{{x = 0.61547971 + 2*pi*n}}}
{{{x = -0.61547971 + 2*pi*n}}}
{{{x = 2.52611294 + 2*pi*n}}}
{{{x = 3.75707236 + 2*pi*n}}}
NOTE 1: The {{{2*pi*n}}} term is a mathematical way of saying "all other coterminal angles."
NOTE 2: The "n" in the above equations can be any integer. By replacing the "n" with various integers we get specific solutions to the original equation. Since there are an infinite number of integers, this term the the "n" is a way to express (and generate) the infinite solutions that exist.
NOTE 3: Some textbooks/teachers use a different letter than "n", like "k". The letter used makes no difference. What matters is that the letter represents "any integer".
2.5) Solve for the variable. Each of these equations is already solved for the variable.<br>
3. Find specific solutions, if requested. Your problem, as posted, does not ask for a specific set of solutions so the solution is the general solution:
{{{x = 0.61547971 + 2*pi*n}}}
{{{x = -0.61547971 + 2*pi*n}}}
{{{x = 2.52611294 + 2*pi*n}}}
{{{x = 3.75707236 + 2*pi*n}}}