given that tan 75 2+root3,without using a table or calculator, determine 5tan15 leaving your answer in the form a+b root c
To find 5tan(15), we can use the half-angle identity for tangent:
tan(x/2) = sqrt((1-cos(x))/(1+cos(x)))
Since x = 30 degrees, we have:
tan(15) = sqrt((1-cos(30))/(1+cos(30)))
To simplify, let's use the trigonometric identities:
cos(30) = sqrt(3)/2
Now we can substitute this value into our equation:
tan(15) = sqrt((1 - sqrt(3)/2)/(1 + sqrt(3)/2))
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator:
tan(15) = sqrt((1 - sqrt(3)/2)/(1 + sqrt(3)/2)) * (sqrt(3)/sqrt(3))
Simplifying, we have:
tan(15) = sqrt((sqrt(3) - 3)/(sqrt(3) + 3))
To get the answer in the form a + b√c, we can multiply the numerator and denominator by the conjugate of the numerator:
tan(15) = (sqrt(3) - 3)/(sqrt(3) + 3) * (sqrt(3) - 3)/(sqrt(3) - 3)
Expanding and simplifying, we have:
tan(15) = (3 - 2sqrt(3))
So, 5tan(15) = 5(3 - 2sqrt(3)) = 15 - 10sqrt(3)
Thus, the answer is 15 - 10√3.
To find the value of 5tan15 without using a table or calculator, we can make use of the trigonometric identity:
tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
Since tan(75°) = 2 + √3, we can rewrite it as:
tan(45° + 30°) = (tan(45°) + tan(30°)) / (1 - tan(45°)tan(30°))
The values of tan(45°) and tan(30°) are known to be 1 and 1/√3, respectively. By substituting the values, we can solve for tan(45° + 30°):
tan(75°) = (1 + 1/√3) / (1 - 1/√3)
To simplify the expression, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:
tan(75°) = ((1 + 1/√3) / (1 - 1/√3)) * (√3 / √3)
Simplifying further:
tan(75°) = (√3 + 1) / (√3 - 1) * (√3 / √3)
tan(75°) = (3 + √3) / (3 - √3)
Now, we can find the value of 5tan15 by using the identity:
5tan15 = 5 * tan(75°)
Substituting the value of tan(75°) that we found earlier:
5tan15 = 5 * (3 + √3) / (3 - √3)
To eliminate the surd in the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator:
5tan15 = (5 * (3 + √3) / (3 - √3)) * ((3 + √3) / (3 + √3))
Simplifying the expression:
5tan15 = (5 * (3 + √3) * (3 + √3)) / ((3 - √3) * (3 + √3))
5tan15 = (5 * (9 + 3√3 + 3√3 + 3)) / (9 - 3)
5tan15 = (5 * (12 + 6√3)) / 6
Simplifying further:
5tan15 = 10 + 5√3
So, the value of 5tan15 in the form a + b√c is 10 + 5√3.