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Oct 28, · If a trig equation can be solved analytically, these steps will do it: Put the equation in terms of one function of one angle. Write the equation as one trig function of an angle equals a constant. Write down the possible value(s) for the angle. If necessary, solve for the variable. Apply any restrictions on the solution. 2 cos(x) = sqrt[3]: x= 30°, °. Putting these the two solution sets together, I get the solution for the original equation as being: x= 30°, 90°, °, °. Solve sin2(?) – sin(?) = 2on the interval 0 ≤ ?.

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General Solution of Trigonometric Equations Trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles. It is expressed as ratios of sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (cosec) angles. For example, cos 2 x + 5 sin x = 0 is a trigonometric equation. Jan 21, · The main objective, as in solving any equation, is to isolate our trig function, using known algebraic methods such as addition, subtraction, factoring, etc., and then ask ourselves one very important question “What angle on the unit circle yields this length or coordinate?”.

Solving trig equations use both the reference angles and trigonometric identities that you've memorized, together with a lot of the algebra you've learned. Be prepared to need to think in order to solve these equations.

In what follows, it is assumed that you have a good grasp of the trig-ratio values in the first quadrant , how the unit circle works, the relationship between radians and degrees , and what the various trig functions' curves look like, at least on the first period.

If you're not sure of yourself, go back and review those topics first. Just as with linear equations , I'll first isolate the variable-containing term:. Note: The instructions gave me the interval in terms of degrees, which means that I'm supposed to give my answer in degrees.

I can see this when I slow down and do the steps. My first step is:. Can any square of a tangent, or of any other trig function be negative? So my answer is:. The left-hand side of this equation factors. I'm used to doing simple factoring like this:. The same sort of thing works here. To solve the equation they've given me, I will start with the factoring:.

I've done the algebra; that is, I've done the factoring and then I've solved each of the two factor-related equations. This created two trig equations. So now I can do the trig; namely, solving those two resulting trigonometric equations, using what I've memorized about the cosine wave. From the first equation, I get:. Putting these the two solution sets together, I get the solution for the original equation as being:.

This equation is "a quadratic in sine"; that is, the form of the equation is the quadratic-equation format:. Since this is quadratic in form, I can apply some quadratic-equation methods. In the case of this equation, I can factor the quadratic:. But the sine is never more than 1 , so this equation is not solvable; it has no solution.

So my complete solution is:. I can use a double-angle identity on the right-hand side, and rearrange and simplify; then I'll factor:.

So the complete solution is:. I'm really not seeing anything here. It sure would have been nice if one of these trig expressions were squared From the last line above, either sine is zero or else cosine is zero, so my solution appears to be:. However and this is important! If you square something, you can't just square-root to get back to what you'd started with, because the squaring may have changed a sign somewhere.

So, to be sure of my results, I need to check my answers in the original equation, to make sure that I didn't accidentally create solutions that don't actually count. Plugging back in, I see:. It's a good thing that I checked my solutions, because two of them don't actually work. They were created by the process of squaring. The answer would have been the same, but I would have needed to account for the solution interval:. After doing the necessary check because of the squaring and discarding the extraneous solutions, my final answer would have been the same as previously.

The squaring trick in the last example above doesn't come up often, but if nothing else is working, it might be worth a try. Keep it in mind for the next test. Page 1 Page 2. All right reserved. Web Design by. Skip to main content. Purplemath Solving trig equations use both the reference angles and trigonometric identities that you've memorized, together with a lot of the algebra you've learned. Content Continues Below. Well, why don't I square both sides, and see what happens?

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