Chapter 8: Q. 56 (page 537)

For any triangle, show that $\cos \frac{C}{2} = \sqrt{\frac{s (s - c)}{a b}}$ where $s = \frac{1}{2} (a + b + c)$

Short Answer

Expert verified

The result $\cos \frac{C}{2} = \sqrt{\frac{s (s - c)}{a b}}$ has been shown.

Step by step solution

Step 1. Given Information

$\cos \frac{C}{2} = \sqrt{\frac{s (s - c)}{a b}}$

Step 2. Consider a triangle

Let us consider a triangle having sides $a, b, c$ and the corresponding opposite angles $A, B, C$ respectively.

By the law of cosines,

$\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$

Step 2. Derive for cos θ

$\cos 2 θ = 2 \cos^{2} θ - 1 \cos^{2} θ = \frac{1 + \cos 2 θ}{2} \cos θ = \sqrt{\frac{1 + \cos 2 θ}{2}}$

Step 4. Derive for cos C2

From the above expression, we get,

$\cos \frac{C}{2} = \sqrt{\frac{1 + \cos C}{2}}$

From the first expression, we get,

$\cos \frac{C}{2} = \sqrt{\frac{1 + \frac{a^{2} + b^{2} - c^{2}}{2 a b}}{2}} = \sqrt{\frac{2 a b + a^{2} + b^{2} - c^{2}}{4 a b}} = \sqrt{\frac{a^{2} + b^{2} + 2 a b - c^{2}}{4 a b}} = \sqrt{\frac{{(a + b)}^{2} - c^{2}}{4 a b}}$

Step 5. Solve the equation

Since

$a + b + c (\frac{1}{2}) = s 2 s - c = (a + b)$

By substituting this value in the above equation,

$\cos \frac{C}{2} = \sqrt{\frac{{(2 s - c)}^{2} - c^{2}}{4 a b}} = \sqrt{\frac{4 s^{2} - 4 s c + c^{2} - c^{2}}{4 a b}} = \sqrt{\frac{s (s - c)}{a b}}$

Hence the required result is proved.

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For any triangle, show that $\cos \frac{C}{2} = \sqrt{\frac{s (s - c)}{a b}}$ where $s = \frac{1}{2} (a + b + c)$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Consider a triangle

Step 2. Derive for cos θ

Step 4. Derive for cos C2

Step 5. Solve the equation

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For any triangle, show that cosC2=s(s-c)abwhere s=12(a+b+c)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Consider a triangle

Step 2. Derive for cos θ

Step 4. Derive for cos C2

Step 5. Solve the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

For any triangle, show that $\cos \frac{C}{2} = \sqrt{\frac{s (s - c)}{a b}}$ where $s = \frac{1}{2} (a + b + c)$