solve the equation 2 tan C-3=3 tan X-4 algebraically for all values of C in the interval odegrees lessthan or equal C lessthan360 degrees.

2tanc - 3 = 3tanc - 4

-3 = tanc - 4

1 = tanc **

Plug in Graphing Calculator:

arctan (1) = 45

45 + 180 = 225

Ans. : 45 and 225.

To solve the equation 2tan(C) - 3 = 3tan(X) - 4 algebraically, and find all values of C in the interval 0° ≤ C < 360°, we can follow these steps:

Step 1: Move all terms involving tan(C) to one side of the equation and all terms involving tan(X) to the other side:
2tan(C) - 3 - 3tan(X) + 4 = 0

Step 2: Combine like terms:
2tan(C) - 3tan(X) + 1 = 0

Step 3: Simplify by factoring out a common factor:
tan(C)(2 - 3tan(X)) + 1 = 0

Step 4: Set each factor equal to zero:
tan(C) = 0 OR 2 - 3tan(X) = 0

Step 5: Solve the first equation, tan(C) = 0. Since we are looking for solutions in the interval 0° ≤ C < 360°, we need to find all angles whose tangent is zero. The tangent function is zero when the angle is a multiple of 180° (or π radians). Therefore, the possible values for C are:
C = 180°k, where k is an integer.

Step 6: Solve the second equation for tan(X):
2 - 3tan(X) = 0

Step 7: Move the constant term to the other side:
3tan(X) = 2

Step 8: Divide both sides by 3:
tan(X) = 2/3

Step 9: We can take the inverse tangent of both sides to solve for X:
X = arctan(2/3) ≈ 33.69°

Step 10: Now, we need to find the values of C for which the tangent of C is equal to the tangent of X, which is 2/3. We can use the periodicity of the tangent function to find all possible values of C by adding multiples of 180° (or π radians) to the reference angle (33.69°). Therefore, the possible values for C are:
C = arctan(2/3) + 180°k, where k is an integer.

This gives us all the solutions for C in the given interval.