Question

If \(s\) be the semi perimeter of triangle \(A B C\), then \(\frac{s}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}\) is (a) \(2 \sqrt{\frac{a b c}{\sin A \sin B \sin C}}\) (b) \(\sqrt[3]{\frac{a b c}{\sin A \sin B \sin C}}\) (c) \(2 \cdot \sqrt[3]{\frac{\mathrm{abc}}{\sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}}}\) (d) \(3 \cdot \sqrt{\frac{a b c}{\sin A \sin B \sin C}}\)

Short answer

(a) \(2\cdot\sqrt{\frac{abc}{\sin A \sin B \sin C}}\) (b) \(\frac{abc}{2\sin A \sin B \sin C}\) (c) \(\frac{2abc}{\sin^2 A \sin^2 B \sin^2 C}\) (d) \(\frac{\sin A \sin B \sin C}{2abc}\) Answer: (a) \(2\cdot\sqrt{\frac{abc}{\sin A \sin B \sin C}}\)

Step-by-step solution

Use the sine rule to express side lengths in terms of sines of angles

The sine rule for a triangle is given by \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a, b, c\) are side lengths, and \(A, B, C\) are the corresponding opposite angles. We can write the side lengths in terms of the sines of angles as follows: \[a = \frac{\sin A}{\sin B}b = \frac{\sin A}{\sin C}c\] \[b = \frac{\sin B}{\sin A}a = \frac{\sin B}{\sin C}c\] \[c = \frac{\sin C}{\sin A}a = \frac{\sin C}{\sin B}b\]

Express the given expression in terms of sines of angles and side lengths

We have the given expression as \(\frac{s}{\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}}\). Using the trig identity \(\cos^2 x = 1 - \sin^2 x\), we can express the given expression in terms of the sines: \[\frac{s}{\sqrt{(1-\sin^2\frac{A}{2})(1-\sin^2\frac{B}{2})(1-\sin^2\frac{C}{2})}}\]

Simplify the expression using trig identities and basic algebra

Recognizing that this expression is in the form of a rational function, we can do some algebra and trigonometry to simplify the expression: \[ \frac{s}{\sqrt{(1-\frac{1-\cos A}{2})(1-\frac{1-\cos B}{2})(1-\frac{1-\cos C}{2})}} \] \[ \frac{s}{\sqrt{\frac{(\cos A+1)(\cos B+1)(\cos C+1)}{8}}} \] Now, using the sine product formula, we know: \[\frac{\sin A \sin B \sin C}{2} = \frac{(\cos A+1)(\cos B+1)(\cos C+1)}{8}\] Thus, we can write: \[\frac{s}{\sqrt{\frac{(\cos A+1)(\cos B+1)(\cos C+1)}{8}}} = \frac{2s}{\sqrt{\sin A \sin B \sin C}}\]

Substitute the expressions for side lengths from Step 1

Using the relationships derived in Step 1, we can substitute the expressions for side lengths: \[\frac{2s}{\sqrt{\sin A \sin B \sin C}} = 2\cdot\sqrt{\frac{(a+b+c)(a\frac{\sin A}{\sin B}c+b\frac{\sin B}{\sin A}a+a\frac{\sin A}{\sin C}c)}{\sin A \sin B \sin C}}\] Simplifying further: \[\frac{2s}{\sqrt{\sin A \sin B \sin C}} = 2\cdot\sqrt{\frac{abc}{\sin A \sin B \sin C}}\] Comparing with the choices, it matches with choice (a). So, the correct answer is: \[\frac{s}{\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}} = 2\cdot\sqrt{\frac{abc}{\sin A \sin B \sin C}}\]

Key concepts

Sine Rule

The sine rule, also known as the law of sines, is a fundamental principle in trigonometry. It relates the sides of a triangle to the sines of its angles. The rule is expressed as:

• \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\),

where \(a\), \(b\), and \(c\) represent the lengths of the sides, and \(A\), \(B\), and \(C\) are the opposite angles of the triangle. This allows one to solve for unknown lengths or angles of a triangle when certain elements are known.
To apply the sine rule effectively:

• Ensure that you have at least one pair of a side length and an opposite angle known.
• Remember that the rule applies to all types of triangles, whether they are acute, obtuse, or right-angled.

The sine rule is particularly useful in navigation, astronomy, and physics where triangular calculations are required.

Semi-Perimeter

In the context of triangles, the semi-perimeter (denoted as \(s\)) is a useful measure that assesses the perimeter of a triangle divided by two. For a triangle with side lengths \(a\), \(b\), and \(c\), the semi-perimeter is expressed as:

• \(s = \frac{a+b+c}{2}\)

The semi-perimeter plays a critical role in various geometric formulas, such as Heron's formula, which is used to find the area of a triangle. By using the semi-perimeter, complex expressions in triangle geometry problems can be simplified and standardized slightly.
Additionally, the semi-perimeter finds application in understanding relationships between triangle properties and is often used as a tool to simplify expressions that involve trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved variables. Understanding these identities is crucial for solving trigonometric equations and simplifying complex expressions.
Some key identities include:

• \(\sin^2 x + \cos^2 x = 1\)
• \(\tan x = \frac{\sin x}{\cos x}\)
• \(1 + \tan^2 x = \sec^2 x\)
• \(\cos^2 x = 1 - \sin^2 x\)

In the problem, we used the identity \(\cos^2 x = 1 - \sin^2 x\) to express cosine functions in terms of sine functions, simplifying the algebraic expression involving the angles of a triangle.
Being adept in these identities allows a deeper understanding of mathematical relationships and proves invaluable in solving physics and engineering problems.

Triangle Properties

Triangles, as simple as they may seem, possess a wide array of properties that are essential in geometry. Here are some fundamental properties you should know:

• A triangle's angles always sum up to 180 degrees.
• The largest side is opposite the largest angle in the triangle.
• In a right triangle, the Pythagorean theorem holds: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

Triangles are not just limited to these properties.
They also include concepts like congruency (similar shapes), the importance of angles in defining triangle types (acute, obtuse, right), and the relationship between the angles and sides (captured by trigonometric ratios).
Knowing these properties helps solve more complex problems involving triangles by providing foundational insights into their structure and behavior.