Working Through Trigonometric Equations on That Worksheet
If you're looking at problems like sin²x + sinx - 2 = 0 or 2cos²x - cosx - 1 = 0 from section 7-5, you're dealing with quadratic-form equations in disguise. They look intimidating until you treat the trig function as a variable and factor it like any other quadratic. The answers themselves aren't hard to find, but missing steps is where most people lose points. Here's what actually works when you're stuck. Take 2sin² - sin - 1 = 0 as an example. You substitute u for sin, giving you 2u² - u - 1 = 0, which factors to (2u + 1)(u - 1) = 0. That means sin = -1/2 or sin = 1. Then you find the angles using your unit circle. For sin = 1, = /2 + 2n. For sin = -1/2, = 7/6 + 2n and = 11/6 + 2n. That's it. The method is the same across every problem in that section. I remember working through a stack of these back when I was tutoring intro pre-calc, and one student kept getting tripped up by a problem where the equation was 4cos²x - 3 = 0. She tried to factor it like a difference of squares and wasted twenty minutes going nowhere. The fix was just isolating cos²x first, getting cos²x = 3/4, then taking the square root to get cosx = ±3/2. Two sets of angles came out of that instead of one. That's the kind of move you have to see before you're comfortable with this material.
Another thing nobody emphasizes enough: when you're solving equations like tan²x - tanx = 0, factoring gives you tanx(tanx - 1) = 0. Setting each factor equal to zero works here because we're dealing with products. But don't fall into the habit of dividing both sides by tanx to simplify. You'll lose the solution where tanx = 0. I've watched that mistake cost people points on exams repeatedly. Always factor, never divide. The answer key you're probably looking for generally covers problems in this range. Basic ones ask for solutions over [0, 2). Harder ones extend to [0, 4) or ask for the general solution. The answers typically look like: Problem 1: = /6, 5/6 (for sin = 1/2 type equations)
Problem 3: = /3, 2/3, 4/3, 5/3 (for cos = -1/2 type equations) Problem 5: = /4, 3/4, 5/4, 7/4 (for tan = 1 type equations) Where things get messy is when you have coefficients in front. Like 3sin²x = 3cos²x. You can't just cancel and say sinx = cosx. First divide both sides by 3, then rearrange to sin²x - cos²x = 0, which is actually -cos(2x) = 0 using the double angle identity. That gives you 2x = /2 + n, so x = /4 + n/2. If you skip the identity step and just take the square root of both sides, you'll get the wrong answers. Trust me, I've seen it happen.
Get the Full Details
Sometimes the answer key won't match your work even when you followed the steps correctly. This usually means the key is listing answers in a different interval or using degrees instead of radians. I've had people panic over this before realizing their answer was equivalent, just written differently. 7/6 and -5/6 are the same angle. If your teacher hasn't specified the interval, write your answer in [0, 2) and move on. For the equations that resist factoring entirely—like sin²x + 2sinx + 2 = 0—you apply the quadratic formula and discover there are no real solutions because the discriminant is negative. The answer key will just say "no solution" for those. Don't force an answer where one doesn't exist. That's a perfectly valid result. The section usually wraps up with reciprocal function equations, which is where cosecant and secant show up. These reduce to sine and cosine once you take reciprocals, so the method doesn't change. But students often forget that cscx = 1/sinx, so they try to solve cscx = 2 by treating it like a direct value. It's the same as sinx = 1/2. Know your reciprocal identities cold or you'll be rewriting everything in terms of sin and cos anyway.
If you're downloading an answer key, make sure it matches your textbook edition. Glencoe's 2005 edition and the newer versions sometimes reorder problems, which makes comparing answers frustrating if you're pulling from the wrong PDF. I found this out the hard way during office hours when three students came in complaining their answers didn't match the key, and they were all using different editions. One last thing about these problems: always check your answers by plugging them back into the original equation. It takes thirty seconds and catches calculation errors before they become grade errors. I used to skip this step and lost points on quizzes for simple arithmetic mistakes that a quick check would have caught immediately.
