Find one value of θ or x that satisfies each of the follo...

Question

Find one value of θ or x that satisfies each of the following.

cot(θ - 10°) = tan(2θ - 20°)

Accepted Answer

Everyone in this problem, we're asked which of the following values of theta satisfies the equation we're given. And the equation is cotangent of three, theta minus 12 degrees is equal to tangent of five, theta minus 26 degrees. We have four answer choices all in degrees. Option A 34 option B 56 option C 12 and option D 16. So let's start and we are gonna write out that equation we're given. So we have code tangent of three, the minus 12 degrees is equal to tangent of phi theta minus 26 degrees. Now, right now, it's a little bit difficult to figure out how we can solve this. We have something in cotangent. We have something in terms of tangent. But what we wanna do is try to get these in terms of the same trigonometric function. So let's recall a relationship we have between code tangent and tangent. OK. And that is that cotangent of 90 degrees minus X is equal to a tangent of X. So in this problem, OK, we're gonna let our X value be what we have inside of our tangent. So five the minus 26 degrees then we're gonna be able to write the right hand side tangent in terms of cotangent. So what we end up with is cotangent of three, theta minus 12 degrees is gonna be equal to cotangent of what? Well, of 90 degrees minus X, which is going to be that value five theta minus 26 degrees and all of that is in brackets. So the negative is gonna apply to both of those terms and that's really important not to mix up those signs. Hey, so now we have cotangent of something is equal to cotangent of something else. OK? So those some must be equal. OK? So we must have that three theta minus 12 degrees is equal to 90 degrees minus phi theta plus 26 degrees. Now, when we write this equality, we have to be a little bit careful. OK. In most cases, we wanna think about all of the solutions. OK? And we know that cotangent has a period of pi so we can shift these solutions by pi in either direction and they're going to get the same result. OK. So we would need to add pi K to one side of our equation RK is an integer in order to find all of the solutions. Now, in this case, we're just looking for a single solution, which one of these satisfies this equation. So we can actually eliminate that pi K when we're looking for just one solution. Now we have a simple equation we're gonna solve for theta. So we're gonna move five theta to the left. We get eight, the, we're gonna move all of the constant terms to the right hand side. We have 90 degrees plus 26 degrees plus 12 degrees. And that gives us 128 degrees. And we can divide both sides by eight. And we get that data is going to be equal to 16 degrees. And so the solution that satisfies or the value that satisfies this equation is going to be option D 16 degrees. Thanks everyone for watching. I hope this video helped see the next one.

Find one value of θ or x that satisfies each of the following.
cot(θ - 10°) = tan(2θ - 20°)

Key Concepts

Relationship Between Cotangent and Tangent

Trigonometric Equation Solving Techniques

Angle Identities and Complementary Angles

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