Consider the following expression: (tan((25 pi)/(24))-tan((3pi)/(8)))/(1+tan((25 pi)/(24))tan((3pi)/(8))) Find the exact value of the expression without a calculator. [Hint: This diagram of special trigonometry values may help. . . Choose 1 answer: (A) -(sqrt3)/(3) (B) (sqrt3)/(3) (C) sqrt3 (D) -sqrt3

Question

Consider the following expression:  (tan((25 pi)/(24))-tan((3pi)/(8)))/(1+tan((25 pi)/(24))tan((3pi)/(8))) Find the exact value of the expression without a calculator. [Hint: This diagram of special trigonometry values may help. . . Choose 1 answer: (A)  -(sqrt3)/(3) (B)  (sqrt3)/(3) (C)  sqrt3 (D)  -sqrt3

Bytelearn · Accepted Answer

Simplify angles in tangent functions: First, let's simplify the angles in the tangent functions. We know that (25 pi)/24 and (3 pi)/8 can be simplified to common trigonometric angles. (25 pi)/24 = (pi)/24 + pi = pi + (pi)/24 (3 pi)/8 = pi/2 - (pi)/8 = (4 pi)/8 - (pi)/8 = (3 pi)/8Use tangent addition formula: Now, let's use the tangent addition formula: tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b)). We can see that our expression has the same structure as the right side of this formula.Rewrite angles in terms of pi: We can rewrite the angles in terms of pi to find their exact values on the unit circle. The angle (25 pi)/24 is equivalent to pi + (pi)/24, which is in the second quadrant where tangent is negative. The angle (3 pi)/8 is in the first quadrant where tangent is positive.Simplify angle inside tangent function: Using the tangent addition formula, we can rewrite the expression as tan((25 pi)/24 - (3 pi)/8). Since (25 pi)/24 is pi + (pi)/24, and (3 pi)/8 is (3 pi)/8, we can simplify this to tan(pi + (pi)/24 - (3 pi)/8).Find exact value of tan(11pi/12): Simplify the angle inside the tangent function: pi + (pi)/24 - (3 pi)/8 = pi + (pi)/24 - (3 pi)/24 = pi - (2 pi)/24 = pi - (pi)/12.Use angle subtraction formula for tangent: Now we have tan(pi - (pi)/12). Since tangent has a period of pi, tan(pi - (pi)/12) = tan(-(pi)/12). This is the same as tan(11pi/12), which is in the second quadrant where tangent is negative.Find exact value of tan(pi/12): We need to find the exact value of tan(11pi/12). We can use the angle subtraction formula for tangent again: tan(11pi/12) = tan(pi - pi/12) = tan(pi) - tan(pi/12) / (1 + tan(pi)tan(pi/12)). Since tan(pi) = 0, this simplifies to -tan(pi/12).Simplify expression for tan(pi/12): To find tan(pi/12), we can use the half-angle formula: tan(pi/12) = tan((pi/6)/2). The half-angle formula for tangent is tan(x/2) = (1 - cos(x)) / sin(x). We need to find the values of sin(pi/6) and cos(pi/6).Simplify expression for -tan(pi/12): The values of sin(pi/6) and cos(pi/6) are 1/2 and sqrt(3)/2, respectively. Plugging these into the half-angle formula, we get tan(pi/12) = (1 - sqrt(3)/2) / (1/2).Correct simplification of -tan(pi/12): Simplify the expression for tan(pi/12): tan(pi/12) = (1 - sqrt(3)/2) / (1/2) = 2 - sqrt(3).Find exact value of original expression: Now we have -tan(pi/12), which is - (2 - sqrt(3)). This simplifies to sqrt(3) - 2.Identify error in previous steps: The original expression simplifies to tan(11pi/12), which we found to be sqrt(3) - 2. Therefore, the exact value of the original expression is sqrt(3) - 2.However, we made a mistake in the simplification process. The correct simplification of -tan(pi/12) should be - (2 - sqrt(3)), which is -2 + sqrt(3). This is not one of the answer choices, indicating an error in the previous steps.

Full solution

More problems from Domain and range

Related topics