Final answer:
The exact value of tan 15 degrees can be found using the tangent subtraction formula and is (2 - √3), which is expressed in simplest radical form.
Step-by-step explanation:
The exact value of tan 15 degrees can be found using a combination of angle sum identities and exact values for more commonly known angles such as 30 and 45 degrees. To find tan 15 degrees, we can express it as tan(45 - 30) degrees and then use the tan subtraction formula: tan(A - B) = (tan A - tan B) / (1 + tan A * tan B).
The exact value of tan 15 degrees can be found using trigonometric identities. We can use the half-angle identity for tangent, which states that tan (x/2) = sqrt((1-cos(x))/(1+cos(x))).
In this case, x is 30 degrees (twice the given angle of 15 degrees). Plugging in the values, we have tan (30/2) = sqrt((1-cos(30))/(1+cos(30))) = sqrt((1-√3/2)/(1+√3/2)) = sqrt((2-√3)/(2+√3)). Rationalizing the denominator, the exact value of tan 15 degrees is (√(2-√3))/(√(2+√3)).
Since the exact values for tan 45 degrees and tan 30 degrees are 1 and 1/√3 respectively, the equation becomes:
- tan 15 degrees = (tan 45 degrees - tan 30 degrees) / (1 + tan 45 degrees * tan 30 degrees)
- tan 15 degrees = (1 - 1/√3) / (1 + 1/√3)
- tan 15 degrees = (√3 - 1) / (√3 + 1)
- tan 15 degrees = (2 - √3)
This is the exact value for tan 15 degrees, expressed in simplest radical form.