Find the acute angle solution to the following equation involving cofunctions. M is in radians. tan ( M 2 + 5 ) = cot ( M − 5 ) \tan\left(\frac{M}{2}+5\right)=\cot\left(M-5\right) tan ( 2 M + 5 ) = cot ( M − 5 )

Let's try solving this problem. So here we are told to find the acute angle solution to the following equation involving cofunctions, and we're told that m is in radians. So given this equation below, what we're trying to do is solve for m. Now I noticed that we have a tangent on the left side of the equation and a cotangent on the right side. This means that we are not dealing with the same trigonometric operations, so we're gonna have to use cofunction identities to solve this problem. Now what I'm first going to start by doing is rewriting the right side of this equation, because cotangent is the the cofunction for this would be just tangent. And when dealing with the cofunction, what we need to do is find the complementary angle. So what I'm gonna do is take 90 degrees and subtract off the angle that we have. Although, this is where you have to be careful, because 90 degrees is what we've done in the past, but notice we're asked for radians. So the radian equivalent of 90 degrees would be pi over 2. So this is something you have to watch out for when solving these types of problems. So this is gonna be pi over 2 minuteus the expression we started with, which is m minus 5, and that's what we get on the right side. And on the left side, this tangent of m over 2+5 is going to remain the same. So this is what our equation looks like. Now from here, what I can do is simplify the right side of this equation. So I can take this minus sign here and distribute it into the parenthesis. So we're going to have the tangent of pi over 2, and then this is going to be minus m, and then this is gonna be these negative signs will cancel, so we'll have plus 5. And then what I can do from here is I can combine this 5 and this pi over 2. Because pi over 2 plus 5, that's the same thing as pi over 2 plus 5 times 2 over 2, and 5 times 2 is 10. So we can write this as 10 over 2, and this is gonna be minus m. Now at this point, what I can do is I can combine these two fractions by adding the numerators. So we're going to have the tangent of 10+pi over 2 minuteusm, and then this whole thing is going to be equal to this left side of the equation, which is the tangent of m over 2 plus 5. This left side of the equation doesn't change because we didn't do any cofunctions on this left side. But notice now that we have the equation written like this, we have the same trigonometric operation being done to both sides. Meaning, we can set the inside of the trigonometric functions equal to each other. So I can set this m over 2 plus 5 equal to what we have on this right side, which is 10+piover2minusm. Now from here, what I can do is subtract this 5 on both sides, because at this point, it's just gonna be algebra that we need to solve for m. So I'll subtract this 5, and we can cancel the fives on the left side, leaving us with just m over 2. And then this is all going to be equal to 10 +2, and then this is minus 5 minuteus m. Now we already discussed how minus 5 is the same thing as minus 5 times 2 over 2, which is just the same thing as 10 over 2. So this is what we get, and then this is all minus m. And so what we can do is combine these fractions together. So we'll have m over 2 on the left side, and this is gonna be equal to 10+pi-ten over 2. Well, this minus 10 and this plus 10 is just gonna cancel each other, so all we're gonna have is pi over 2, because we have this common denominator, minus m. So this is where we end up. And from here, I wanna add this m to both sides, because I'm trying to get m by itself. So I'll add m on the left and right side. This is going to get the m's to cancel on the right side, leaving us with just pi over 2 on the right. And then on the left side, we're going to have m plus m over 2. And so m over 2, plus m is the same thing as 2m over 2, and 2m plus m is 3m, so we're going to have 3m over 2 equals pi over 2. Now to solve for m, well, what I can first do is multiply both sides of the equation by 2, and that's gonna get the 2's to cancel completely in this equation, leaving us with just 3m is equal to pi. And then from here, I just need to divide this 3 on both sides of the equation, getting the 3's to cancel on the left, giving us that m is equal to pi over 3. So pi over 3 radians is the solution to this problem. Hope you found this video helpful. Thanks for watching.