Introduction / Context:
This is a trigonometry question that uses the relationship between complementary angles in a right triangle. The problem combines standard angle values with simple identities to obtain a neat simplified result.
Given Data / Assumptions:
- Triangle ΔABC is right angled at B, so ∠B = 90°.
- ∠A = 60°.
- The third angle ∠C is therefore 30°.
- We must compute Sec C * Sin A.
Concept / Approach:
In a right triangle, the two acute angles are complementary and add up to 90°. So if one is 60°, the other must be 30°. We use standard trigonometric values: Sin 60° and Cos 30°. Sec is the reciprocal of Cos, so Sec C = 1 / Cos C. Then we multiply Sec C by Sin A and simplify.
Step-by-Step Solution:
Given ∠A = 60° and ∠B = 90°, the remaining angle is ∠C = 180° − 90° − 60° = 30°.
We need Sec C * Sin A = Sec 30° * Sin 60°.
Recall that Cos 30° = √3 / 2, so Sec 30° = 1 / Cos 30° = 2 / √3.
Also recall that Sin 60° = √3 / 2.
Now compute Sec 30° * Sin 60° = (2 / √3) * (√3 / 2).
The numerator becomes 2 * √3 and the denominator becomes √3 * 2, so the factors cancel.
Thus Sec 30° * Sin 60° = 1.
Verification / Alternative check:
Using approximate decimals, Sin 60° is about 0.866 and Cos 30° is also about 0.866. Hence Sec 30° is about 1 / 0.866 which is roughly 1.155. The product 1.155 * 0.866 is close to 1, confirming our exact symbolic result.
Why Other Options Are Wrong:
Options a, b, c, and e correspond to individual values of common trigonometric ratios but not to the product Sec 30° * Sin 60°. None of them gives exactly 1 when evaluated, so they do not match the simplified expression.
Common Pitfalls:
Some learners mistakenly treat A and C as both 60° or both 30°, forgetting that the sum of angles in a triangle must be 180°. Another error is to confuse Cos 30° with Sin 30° or to misremember the special angle values. Always confirm the angle measures and recall the standard ratio table before substituting values.
Final Answer:
The value of Sec C × Sin A is
1.
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