Chapter 8: Q. 51 (page 543)

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .

Short Answer

Expert verified

It is proved that $\cot \frac{C}{2} = \frac{s - c}{r}$ .

Step by step solution

Step 1. Given Informtion

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

use localid="1650175628630" $r = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{\cos \frac{C}{2}}$

localid="1650175390705" $\sin \frac{C}{2} = \sqrt{\frac{(s - a) (s - b)}{a b}} \cot = \frac{\sin}{\cos}$

Step 2. Proving the statement

$r = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{\cos \frac{C}{2}} \cos \frac{C}{2} = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{r}$

$\frac{\cos \frac{C}{2}}{\sin \frac{C}{2}} = \frac{c}{r} \frac{\sin \frac{A}{2} \sin \frac{B}{2}}{\sin \frac{C}{2}} \cot \frac{C}{2} = \frac{c}{r} \frac{\sin \frac{A}{2} \sin \frac{B}{2}}{\sin \frac{C}{2}}$

Now.

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

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Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .

Short Answer

Step by step solution

Step 1. Given Informtion

Step 2. Proving the statement

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Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show thatcotC2=s-crwheres=12a+b+c.

Short Answer

Step by step solution

Step 1. Given Informtion

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .