The half-angle formula is used to find the value of the trigonometric ratios like 22.5°, 15°. half-angle of trigonometric functions with the help of an angle. These formulae can be derived from the reduction formulas and we can use them when we have an angle that is half the size of a special angle. It is used to find the exact value of the trigonometric ratios of 15 (half of 30 degrees), 22.5( half of 45 degrees), and so on.
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In this article, we will cover the concept of the half-angle formula. This concept 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.
The half-angle formula is used to find the value of the trigonometric ratios of the angles like 22.5° (which is half of the angle 45°), 15° (which is half of the angle 30°), etc.
The half-angle formula can be derived with the help of the reduction formula.
1.
2.
3.
Note that the half-angle formulas are preceded by a
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have
We can also represent the half-angle formula using the side of the triangle.
We know that,
Now, for any
Using the above two formulas
In a similar way, we can derive other formulas.
We know that
Using the above two formulas
As
In a similar way, we can derive other formulas
This half-angle formula can be proved using tan x = sin x/cos x, and using the half-angle formula of sine and cosine.
This half-angle formula can be proved using $\tan x=\sin x / \cos x and using the half-angle formula of sine and cosine.
The half-angle formulas in trigonometry serve as crucial tools for simplifying and relating trigonometric functions of angles halved to those of their originals. These formulas have applications in various fields including mathematics, physics, and engineering. Understanding these formulas helps us to solve theoretical and practical problems.
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Example 1: If
Solution
Hence, the answer is 0
Example 2: The value of
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Solution:
Hence, the answer is
Example 3: If
Solution:
Trigonometric Ratios of Submultiples of an Angle -
This shows the transformation formulae and double angle formulae.
Therefore,
dividing (2) by (1), we get
[By making a triangle]
Thus,
Hence, the answer is
Example 4: Let
Solution:
Squaring and adding,
Example 5: Find the range of functions
Solution:
Hence, the answer is
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