In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP!

Examples on Semiperimeter and Half Angle Formulae

Go back to  'Trigonometry'

Example - 50

Evaluate  \(\begin{align}S = (b - c)\cot \frac{A}{2} + (c - a)\cot \frac{B}{2} + (a - b)\cot \frac{C}{2}\end{align}\)  .

Solution: By the half-angle formula for  \(\begin{align}tan \frac{A}{2},\end{align}\)  we have

\[\begin{align}&\cot \frac{A}{2} = \sqrt {\frac{{s(s - a)}}{{(s - b)(s - c)}}}  \\  \,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\;\;= \frac{{s(s - a)}}{{\sqrt {s(s - a)(s - b)(s - c)} }} = \frac{{s(s - a)}}{\Delta }  \\   \Rightarrow\quad   &\;\;\;\;\;\;S = \frac{{s(s - a)(b - c) + s(s - b)(c - a) + s(s - c)(a - b)}}{\Delta } \\ \,\,\,\,\, &\qquad\;\;= \frac{{s\left( {({b^2} - {c^2}) + ({c^2} - {a^2}) + ({a^2} - {b^2})} \right)}}{\Delta } \\ \,\,\,\,\, &\qquad\;\;= 0 \\ \end{align}\]

Example - 51

Prove that \(\begin{align}\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \left( {\frac{{a + b + c}}{{b + c - a}}} \right)\cot \frac{A}{2}\end{align}\)

Solution:

\[\begin{align}& \text{LHS}=\frac{s(s-a)}{\Delta }+\frac{s(s-b)}{\Delta }+\frac{s(s-c)}{\Delta }\left\{ \begin{gathered}& \text{Using}\ \text{the}\ \text{expression}\ \text{for}\cot \frac{A}{2}\  \\ & \text{from}\ \text{the}\ \text{previous}\ \text{example} \\ \end{gathered} \right\} \\  & \,\,\,\,\,\,\,\,\,\,\,\,=\frac{s(3s-(a+b+c))}{\Delta } \\  &  \,\,\,\,\,\,\,\,\,\,\,\,=\frac{{{s}^{2}}}{\Delta } \\  & \text{RHS}=\left( \frac{a+b+c}{b+c-a} \right)\cot \frac{A}{2} \\  & \,\,\,\,\,\,\,\,\,\,\,\,=\frac{2s}{2s-2a}\cdot \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} \\ & \,\,\,\,\,\,\,\,\,\,\,=\frac{{{s}^{2}}}{\sqrt{s(s-a)(s-b)(s-c)}} \\  & \,\,\,\,\,\,\,\,\,\,\,=\frac{{{s}^{2}}}{\Delta } \\ \end{align}\]

Example - 52

Consider the following statements concerning \(\Delta ABC\)  :

I. The sides a, b, c and \(\Delta \) are rational

II. \(a,\;\tan \frac{B}{2},\;\tan \frac{C}{2}\) are rational

III. \(a,\;\sin A,\;\sin B,\;\sin C\) are rational

Show that I \( \Rightarrow \)  II \( \Rightarrow \) III  \( \Rightarrow \)   I

Solution: Assuming I is true, then  \(\begin{align}s = \frac{{a + b + c}}{2}\end{align}\)  is rational. Since  \(\Delta \) is also rational.

\[\tan \frac{B}{2} = \frac{\Delta }{{s(s - b)}}\;\;{\text{and}}\;\;\tan \frac{C}{2} = \frac{\Delta }{{s(s - c)}}\]

are also rational. Thus, I \( \Rightarrow \)  II.

Now, assuming II, we see that

\[\sin B = \frac{{2\tan \frac{B}{2}}}{{1 + {{\tan }^2}\frac{B}{2}}}\;\;{\text{and}}\;\,\sin C = \frac{{2\tan \frac{C}{2}}}{{1 + {{\tan }^2}\frac{C}{2}}}\]

are rational. Also,

\[\tan \frac{A}{2} = \cot \left( {\frac{{B + C}}{2}} \right) = \frac{{1 - \tan \frac{B}{2}\tan \frac{C}{2}}}{{\tan \frac{B}{2} + \tan \frac{C}{2}}}\] 

becomes rational, which implies that sin A is rational. Thus, II  \(\Rightarrow \)  III

Finally, assuming III, we have

\[b = a\frac{{\sin B}}{{\sin A}}\;\;{\text{and}}\;\;c = a\frac{{\sin C}}{{\sin A}}\]

as rational, and further \(\Delta  = \frac{1}{2}bc\;\sin A\) , as rational. Thus, III  \(\Rightarrow \) I.

Example - 53

Prove that  \({s^2} > 4\Delta \)

Solution: Applying the A.M – G. M. inequality on \(s,\;s - a,\;s - b\;{\text{and}}\;s - c,\)

we have

\[\begin{align}&\frac{{s + (s - a) + (s - b) + (s - c)}}{4} > {\left( {s(s - a)(s - b)(s - c)} \right)^{1/4}}\\   \Rightarrow\quad   &\frac{s}{2} > \sqrt \Delta     \quad\Rightarrow\quad {s^2} > 4\Delta \\ \end{align} \]

Note that equality can never hold since s cannot equal the other terms.

Example - 54

Prove that  \(\begin{align}(abcs)\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2} = {\Delta ^2}\end{align}\)

Solution: Using the half-angle sine formulae should yield the desired result:

\[\begin{align}&{\text{LHS}} = abcs\sqrt {\frac{{(s - b)(s - c)}}{{bc}}} \sqrt {\frac{{(s - c)(s - a)}}{{ca}}} \sqrt {\frac{{(s - a)(s - b)}}{{ab}}}  \\  \,\,\,\,\,\,\,\,\,\,\,\, &\qquad= \frac{{abc\;s(s - a)(s - b)(s - c)}}{{abc}} \\  \,\,\,\,\,\,\,\,\,\,\,\, &\qquad= {\Delta ^2} \\ \end{align} \]

Download SOLVED Practice Questions of Examples on Semiperimeter and Half Angle Formulae for FREE
Trigonometry
grade 11 | Questions Set 1
Trigonometry
grade 11 | Answers Set 1
Trigonometry
grade 11 | Questions Set 2
Trigonometry
grade 11 | Answers Set 2
Download SOLVED Practice Questions of Examples on Semiperimeter and Half Angle Formulae for FREE
Trigonometry
grade 11 | Questions Set 1
Trigonometry
grade 11 | Answers Set 1
Trigonometry
grade 11 | Questions Set 2
Trigonometry
grade 11 | Answers Set 2
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school