alleryStart learning →With tan(pi/6) = 1 / sqrt(3) and tan(pi/4) = 1, find the exact value of tan(pi/6 + pi/4).
A(sqrt(3) - 1) / (sqrt(3) + 1)
B(1 - sqrt(3)) / (sqrt(3) + 1)
C(sqrt(3) + 1) / (sqrt(3) - 1)
D(sqrt(3) + 1) / (1 - sqrt(3))
Answer & Solution
Correct answer: C. (sqrt(3) + 1) / (sqrt(3) - 1)
1. Use tan(a + b) = (tan a + tan b) / (1 - tan a tan b).
2. The numerator is 1 / sqrt(3) + 1, which is (1 + sqrt(3)) / sqrt(3).
3. The product tan a tan b is (1 / sqrt(3))(1) = 1 / sqrt(3).
4. So the denominator is 1 - 1 / sqrt(3), which is (sqrt(3) - 1) / sqrt(3).
5. Dividing, the two factors of sqrt(3) cancel and leave (1 + sqrt(3)) / (sqrt(3) - 1).
6. That is the same as (sqrt(3) + 1) / (sqrt(3) - 1), a positive number of about 3.73, matching a 75 degree angle.
7. The options with 1 - sqrt(3) in the denominator are negative, which cannot describe an angle in the first quadrant.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.2 Sum and Difference Identities_
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