Difficulty: Easy
Correct Answer: 7/3
Explanation:
Introduction / Context:
This problem again tests knowledge of exact trigonometric values for standard angles but in a slightly different combination. It reinforces the ability to work with secant and tangent functions and to simplify expressions with surds and simple fractions.
Given Data / Assumptions:
Concept / Approach:
We recall that cos 45° = √2/2 and sec 45° is its reciprocal. Similarly, tan 30° is sin 30° divided by cos 30°. Once those values are substituted, the expression becomes a sum of two rational numbers and simplifies to a single fraction.
Step-by-Step Solution:
cos 45° = √2/2 so sec 45° = 1 / (√2/2) = 2/√2 = √2
Hence √2 · sec 45° = √2 · √2 = 2
For 30°, sin 30° = 1/2, cos 30° = √3/2
So tan 30° = (1/2) / (√3/2) = 1/√3
Then (1/√3) · tan 30° = (1/√3) · (1/√3) = 1/3
Total expression = 2 + 1/3
2 = 6/3 so 2 + 1/3 = 7/3
Verification / Alternative check:
Approximate numerically: sec 45° ≈ 1.414 and √2 ≈ 1.414 so their product is exactly 2. Tan 30° ≈ 0.577 and 1/√3 ≈ 0.577 so their product is about 0.333. The total is roughly 2.333 which equals 7/3 as a decimal. This numerical check confirms both the logic and the simplification.
Why Other Options Are Wrong:
(1 + √3)/2 and (3 + 2√2)/(3√2) involve surds in more complicated forms and yield different numerical values. (1 + 3√2)/√3 is much larger than 7/3. The value 2 ignores the tan term entirely and is therefore incomplete.
Common Pitfalls:
Errors often come from misremembering tan 30° as √3 or 1/√3 in the wrong place. Some students forget to multiply 1/√3 by tan 30° and instead add them. Another mistake is incorrectly simplifying sec 45° as 1/√2 instead of √2. Careful recall of standard angle values prevents these mistakes.
Final Answer:
So the expression √2 · sec 45° + (1/√3) · tan 30° simplifies to 7/3.
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