In trigonometry, the half-angle formula is used to find the value of angles apart from 30,45,60,90. It makes our calculation with the help of the half-angle formula we can find the value of any angle. In real life, we use the half-angle formula to calculate the angle of the roof, ceramic tile installation, etc.
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In this article, we will cover the concept of the half-angle formula. This category falls under the broader category of Trigonometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of eight questions have been asked on this concept.
The half-angle formula helps us to find the value of angles like
We have the following half-angle formula for sine in triangle ABC,
Now, for any
Using the above two formulas
As
In a similar way, we can derive other formulas.
We have the following half-angle formula for cosine in triangle ABC
Derivation of Half Angle Formula of Cosine
We know that
And for any
Using the above two formulas
Similarly, we can derive other formulas
We have the following half-angle formulas for tan in triangle ABC,
This half-angle formula can be proved using
Half Angle Formula
The half-angle formulas are preceded by a
1.
2.
3.
Derivation of Half Angle Formula of sine
The half-angle formula for sine is derived as follows:
Derivation of Half Angle Formula of Cosine
To derive the half-angle formula for cosine, we have
Derivation of Half Angle Formula of Tangent
Summary: The half-angle formula helps us to find the value of angles like
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Solved Examples Based on Half-Angle Formula
Example 1: In triangle
Solution
Half-Angle Formula (in terms of perimeter and sides of the triangle)
Hence, the answer is sides of triangle
Example 2: If in a triangle
Solution: Half-Angle Formula (in terms of perimeter and sides of the triangle.
Hence, the answer is angle C is
Example 3: The value of
Solution
Hence, the answer is
Example 4: In triangle ABC if
Solution
(since
Hence, the answer is A.P
Example 5: In a
Solution: Trigonometric Ratios of Functions -
- wherein
Hence, the answer is
Frequently Asked Questions (FAQs)
Q1) what is the half-angle formula for cosine?
Answer: We have the following half-angle formula for cosine in triangle ABC
We have the following half-angle formulas for tan in triangle ABC,
$
\begin{aligned}
& \tan \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \\
& \tan \frac{B}{2}=\sqrt{\frac{(s-a)(s-c)}{s(s-b)}} \\
& \tan \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}
\end{aligned}
$
$
\begin{aligned}
& \cos \frac{B}{2}=\sqrt{\frac{a+c}{2 c}} \\
& \sqrt{\frac{s(s-b)}{a c}}=\sqrt{\frac{a+c}{2 c}} \quad \text { where } s=\frac{a+b+c}{2} \\
& \frac{s(s-b)}{a c}=\frac{a+c}{2 c} \\
& 2 s(s-b)=a(a+c) \\
& (a+b+c)(a+c-b)=2 a(a+c) \\
& (a+c)^2-b^2=2 a^2+2 a c \\
& a^2+b^2=c^2 \\
& \angle C=\frac{\pi}{2}
\end{aligned}
$
We have the following half-angle formula for sine in triangle ABC,
$
\begin{aligned}
\sin \frac{A}{2} & =\sqrt{\frac{(s-b)(s-c)}{b c}} \\
\sin \frac{B}{2} & =\sqrt{\frac{(s-a)(s-c)}{a c}} \\
\sin \frac{C}{2} & =\sqrt{\frac{(s-a)(s-b)}{a b}}
\end{aligned}
$
Semi-perimeter of the $\triangle A B C$, is
$
\text { is } s=\frac{a+b+c}{2}
$
and it is denoted by $s$. So, the perimeter of $\triangle A B C$ is $2 s=a+b+c$
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