Concept Check Find a solution for each equation. tan (3θ ― 4°) ...

Question

Concept Check Find a solution for each equation.   tan (3θ ― 4°) = 1 / [cot(5θ ― 8°)]

Accepted Answer

Welcome back everyone. In this problem, we want to determine the value of beta for the equation. The tangent A four beta minus 13 degrees equals one divided by the core tangent of 11 beta minus 27 degrees. For our answer choices A is two degrees. B is negative two degrees C is six and D is nine degrees. Now, what do we want to know here? Well, we're trying to figure out the value of beta and we're working with an equation that has the tangent and the core tangent. And both of those are closely related because we can recall that the core tangent of an angle is the reciprocal of the tangent of that angle. And likewise, if that's true, then it makes sense that the tangent of an angle would be the reciprocal of the core tangent of that angle. So if we can use this idea to help us rewrite one of our expressions with a similar trigonometric function, then it would be easier to work our equation. So let's use this idea, the tangent. It is the reciprocal of the core tangent on the right hand side of our equation. So by doing that then we can rewrite our equation as the tangent of four beta minus 13 degrees equals the tangent of 11 beta minus 27 degrees. Because remember the tangent is the reciprocal of the core tangent. No, we have the tangent on both sides. It will be easier to solve because since both tangents are equal, that must mean their angles are equal. So that means for beta, for some reason whenever I draw beta, I'm tempted to draw really long four, I just can't explain it. But for beta minus 13 degrees equals beta minus 27 degrees. No, we can go ahead and solve for beta. First, let's get rid of beta. On the right hand side by subtracting 11 beta from both sides. Now four beta minus 11 beta equals negative seven beta. And we will rewrite negative 13 degrees, 11 beta minus 11 beta equals zero. So our right hand side will have negative 27 degrees. We can add 13 to both sides which negative 13 plus 13 equals zero. So negative seven beta will be left on the left hand side while on the right hand side, negative 27 plus 13 equals negative 14 degrees. Therefore, if we divide both sides by negative seven, negative seven, divided by negative seven equals one. So we are left with beta and the negative 14 divided by negative seven equals positive two. Therefore beta equals two degrees. When we go back to our answer choices that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

Concept Check Find a solution for each equation. tan (3θ ― 4°) = 1 / [cot(5θ ― 8°)]

Key Concepts

Tangent and Cotangent Functions

Trigonometric Identities

Solving Trigonometric Equations

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