This is a free lesson from our course in Trigonometry
In this lesson you'll go through the basic concepts of double angle and the half angle formulas. Further you'll deal with problems
that involve angular measures and learn it with the help of some examples, practice
questions with solution and using video explanations by the instructor that brings
in an element of real-class room experience.
The double and half angle formulas can be used to find the values of unknown trigonometric
functions. For example, you might not know the sine of 15 degrees, but by using
the half angle formula for sine, you can figure it out based on the common value
of sin(30) = 1/2. They are also useful for certain integration problems where a
double or half angle formula may make things much simpler to solve. The use of Double-Angle
formulas help reduce the degree of difficulty.
(More text below video...)
(Continued from above)
Let us see; how to derive the double angle formulas for sine
and cosine using their sum formulas:
For an angle with measure A, the double angle formula
for sine function is
sin 2A = 2 sin A
cos A.
In the sum formula sin (A + B) = sin A cos B
+ cos A sin B, on changing B with A
in, we'll get double angle formula for sine.
Also the double angle formulas for cosine function are
cos 2A = cos2A
- sin2A
=
1 - 2 sin2A
= 2 cos2A - 1.
Similarly, for an angle with measure A, the half angle formula for sine
and cosine trigonometric functions are
sin (A/2) =
(1
- cos A)/2.
cos (A/2) =
(1
+ cos A)/2.
For example, cos 15
can be evaluated using the formula cos (30/2)
=
(1 + cos 30)/2,
and on substituting cos 30
=
3/2 in this, you'll
get
{(2 +
3)}/2.
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(1
- cos A)/2.
can be evaluated using the formula cos (30