The sides of a triangle are 7 cm, 5 cm, and 3 cm. Then the largest angle of the triangle is:

ESIC Nursing Officer - 2019 (Shift-2)

Non Nursing Subjects

Hard

Appeared in: ESIC Nursing Officer - 2019 (Shift-2)

Explanation

The largest angle in any triangle is always opposite the longest side. In this case, the longest side is 7 cm.
The Law of Cosines is the correct formula to use for finding an angle when all three side lengths are known.
The formula is cos(A) = (b² + c² - a²) / 2bc, where 'a' is the side opposite the angle 'A'.
Substituting the values gives cos(A) = (5² + 3² - 7²) / (2 * 5 * 3), which simplifies to -15 / 30 or -0.5.
The angle whose cosine is -0.5 is 120°. A negative cosine value indicates that the angle is obtuse (greater than 90°).

Option A: The cosine of 45° is approximately 0.707, not -0.5. This would be the angle in a different triangle.
Option B: The cosine of 60° is 0.5, not -0.5. A common mistake is to ignore the negative sign in the calculation.
Option C: The cosine of 36° is approximately 0.809, which does not match the calculated value of -0.5.

Visual 1: Diagram - A triangle labeled with sides 3 cm, 5 cm, and 7 cm. The angle opposite the 7 cm side is marked as 'A' to illustrate it is the largest angle.
Visual 2: Infographic - The Law of Cosines formula, c² = a² + b² - 2ab cos(C), and its rearranged form for finding an angle, cos(C) = (a² + b² - c²) / 2ab.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain This question tests the application of the Law of Cosines in trigonometry to determine the angles of a triangle given the lengths of its three sides as background academic context rather than a clinical decision trigger.
This is a mathematics question testing trigonometry concepts, which is a part of the non-nursing syllabus for some nursing entrance exams.
While not directly used in daily nursing tasks, a strong foundation in mathematics helps in understanding and performing drug dosage calculations, interpreting charts, and analyzing statistical data in research.
What if? - If the question asked for the smallest angle, you would apply the same Law of Cosines but solve for the angle opposite the shortest side (3 cm). In that case, cos(C) = (7² + 5² - 3²) / (2 * 7 * 5) = 65/70 ≈ 0.928, giving an angle of approximately 21.8°.

How to Approach the Question

First, read the question carefully to identify the given information: the lengths of three sides of a triangle (7 cm, 5 cm, 3 cm).
Identify what needs to be found: the largest angle of the triangle.
Recall the geometric principle: the largest angle in a triangle is always opposite the longest side.
Identify the longest side, which is 7 cm.
Select the correct mathematical tool. To find an angle when all three sides are known, the Law of Cosines is used.
Apply the Law of Cosines formula, cos(A) = (b² + c² - a²) / 2bc, setting 'a' as the longest side (7 cm).

Concept Tested & Keywords

Concept Tested: This question tests the application of the Law of Cosines in trigonometry to determine the angles of a triangle given the lengths of its three sides.
Stem keywords: triangle, sides, largest angle
Lead-in keywords: Then the largest angle of the triangle is

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