Find the exact values:
tan(7pi/4) - tan (pi/6)
-1 - 1/√3
To find the exact values of tan(7π/4) - tan(π/6), we first need to calculate the individual tan values and then subtract them.
1. Tan(7π/4):
- Start by converting 7π/4 into degrees. Since 2π radians is equal to 360 degrees, we can find the equivalent degrees for 7π/4.
- Multiply 7π/4 by (180/π) to convert it into degrees. The π in the numerator and denominator will cancel out, leaving us with (7 * 180) / 4 = 315 degrees.
- The equivalent angle for 315 degrees lies in the fourth quadrant, where the tangent value is negative. Therefore, we can determine tan(7π/4) = -1.
2. Tan(π/6):
- Convert π/6 into degrees by multiplying it by (180/π). The π in the numerator and denominator will cancel out, leaving us with (1 * 180) / 6 = 30 degrees.
- The equivalent angle for 30 degrees lies in the first quadrant, where the tangent value is positive. Therefore, we can determine tan(π/6) = 1/√3 = √3 / 3.
3. Calculate tan(7π/4) - tan(π/6):
- Substitute the values of tan(7π/4) and tan(π/6) into the equation: -1 - (√3 / 3).
- To simplify this subtraction, we need to have the same denominator for both fractions.
- Multiply -1 by 3/3 to have a common denominator. The equation becomes -3/3 - (√3 / 3).
- Subtraction of fractions with the same denominator yields: (-3 - √3) / 3.
Therefore, the exact value of tan(7π/4) - tan(π/6) is (-3 - √3) / 3.