Lesson Trigonometric functions of multiply argument - Examples

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Trigonometric functions of multiply argument - Examples


The formulas for the Trigonometric functions of multiply argument are:

cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1,

sin%282alpha%29+=+2%2Asin%28alpha%29%2Acos%28alpha%29,

cos%283alpha%29+=+4%2Acos%5E3%28alpha%29+-+3%2Acos%28alpha%29,

sin%283alpha%29+=+-4%2Asin%5E3%28alpha%29+%2B+3%2Asin%28alpha%29.

The proofs of these formulas are presented in the lesson Trigonometric functions of multiply argument in this module.
Below are examples of applications of these formulas.

Example 1
Find cos(15°), sin(15°), tan(15°).

Solution
First, find cos(15°).
Put alpha = 15°. Note that 2alpha = 30° and use the formula for cosines of the double argument:

cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1.

Substitute alpha = 15°, 2alpha = 30° and cos%282alpha%29 = cos(30°) = sqrt%283%29%2F2 into this formula. You get the equation
sqrt%283%29%2F2 = 2*cos^2(15°) - 1, or

cos^2(15°) = %282%2Bsqrt%283%29%29%2F2.

Hence,
cos(15°) = sqrt%282%2Bsqrt%283%29%29%2F2.

Since you already calculated cos(15°), you can easily find sin(15°):
sin^2(15°) = 1 - cos^2(15°) = 1+-+%282%2Bsqrt%283%29%29%2F4+=+%282-sqrt%283%29%29%2F4,

hence,
sin(15°) = sqrt%282-sqrt%283%29%29%2F2.

Now,
tan(15°) = sin(15°)/cos(15°) = sqrt%282-sqrt%283%29%29%2Fsqrt%282%2Bsqrt%283%29%29.

Note that sin(15°), cos(15°) and tan(15°) were just calculated differently in the lessons
Addition and subtraction formulas - Examples and
Product of trigonometric functions - Examples in this module.
Please make sure that all relevant results from these lessons are identical.

Example 2
Find sin(18°), cos(18°) and tan(18°).

Solution
Let us denote alpha = 18°.
Then 5alpha = 90°,
hence 2alpha = 90°-3alpha.

Therefore,
sin%282alpha%29 = sin%28pi%2F2-3alpha%29,
and consequently
sin%282alpha%29+=+cos%283alpha%29
(which is, actually, the obvious equality sin(36°) = cos(54°)).

Now, apply the formula for the double argument to sines at the left side and the formula for the triple argument to cosines at the right side.

After applying these formulas you get
2%2Asin%28alpha%29%2Acos%28alpha%29+=+4%2Acos%5E3%28alpha%29+-+3%2Acos%28alpha%29.

Since cos%28alpha%29 is not equal to zero, you can divide both sides of the precedent equality by cos%28alpha%29. You get the equation
2%2Asin%28alpha%29+=+4%2Acos%5E2%28alpha%29+-+3.

Now, introduce x=sin%28alpha%29 for short and replace cos%5E2%28alpha%29+=+1-x%5E2 in the precedent formula. You get the equation
2x+=+4%281-x%5E2%29+-3,
or, after simplifying,
4x%5E2+%2B+2x+-+1+=+0.

This is the quadratic equation. Solve it using the quadratic formula (see the lesson Introduction into Quadratic Equations in this site).
You get two roots
, and
.

Only the first root fits (the second root doesn't fit due to its sign).
So, the answer is: sin(18°) = %28-1%2Bsqrt%285%29%29%2F4.

Since you already calculated sin(18°), you can easily find cos(18°):
cos^2(18°) = 1 - sin^2(18°) = .
Hence,
cos(18°) = sqrt%28%285%2Bsqrt%285%29%29%2F8%29.

Consequently,
tan(18°) = sin(18°)/cos(18°) = %28-1%2Bsqrt%285%29%29%2F%28sqrt%282%29%2Asqrt%285%2Bsqrt%285%29%29%29.

Example 3
Find sin(36°), cos(36°) and tan(36°).

Solution

You just learned (from the precedent Example) that cos(18°) = sqrt%28%285%2Bsqrt%285%29%29%2F8%29.
Apply the formula of cosines for the double argument to calculate cos(36°):

cos(36°) = cos(2*18°) = 2*cos^2(18°) - 1 = 2%2A%285+%2B+sqrt%285%29%29%2F8+-+1+=+%281+%2B+sqrt%285%29%29%2F4.

Now, you can easily calculate sin(36°):
sin^2(36°) = 1 - cos^2(36°) = ,
therefore sin(36°) = sqrt%28%285-sqrt%285%29%29%2F8%29.

Consequently
tan(36°) = sin(36°)/cos(36°) = %28sqrt%282%29%2Asqrt%285-sqrt%285%29%29%29%2F%281%2Bsqrt%285%29%29.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For your convenience, below is the list of my lessons on Trigonometry in this site in the logical order.
They all are under the current topic Trigonometry in the section Algebra II.

Addition and subtraction formulas
cos%28alpha+%2B+beta%29+=+cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%29,
cos%28alpha+-+beta%29+=+cos%28alpha%29%2Acos%28beta%29+%2B+sin%28alpha%29%2Asin%28beta%29,
sin%28alpha+%2B+beta%29+=+sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29,
sin%28alpha+-+beta%29+=+sin%28alpha%29%2Acos%28beta%29+-+cos%28alpha%29%2Asin%28beta%29,

, .
    The lessons Addition and subtraction formulas and
                     Addition and subtraction formulas - Examples







Addition and subtraction of trigonometric functions
,

,

,

,

, .
    The lessons Addition and subtraction of trigonometric functions and
                     Addition and subtraction of trigonometric functions - Examples












Product of trigonometric functions
,

,

.
                                 The lessons Product of trigonometric functions and
                                                   Product of trigonometric functions - Examples






Powers of trigonometric functions
cos%5E2%28alpha%29+=+%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

sin%5E2%28alpha%29+=+-%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

cos%5E3%28alpha%29+=+%281%2F4%29%2Acos%283alpha%29+%2B+%283%2F4%29%2Acos%28alpha%29,

sin%5E3%28alpha%29+=+-%281%2F4%29%2Asin%283alpha%29+%2B+%283%2F4%29%2Asin%28alpha%29.
                                          The lessons Powers of Trigonometric functions and
                                                            Powers of Trigonometric functions - Examples









Trigonometric functions of multiply argument
cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1,

sin%282alpha%29+=+2%2Asin%28alpha%29%2Acos%28alpha%29,

cos%283alpha%29+=+4%2Acos%5E3%28alpha%29+-+3%2Acos%28alpha%29,

sin%283alpha%29+=+-4%2Asin%5E3%28alpha%29+%2B+3%2Asin%28alpha%29.
                                                The lessons Trigonometric functions of multiply argument and
                                                                Trigonometric functions of multiply argument - Examples (this lesson)








Trigonometric functions of half argument
sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2, cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

,

sin%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, cos%28alpha%29+=+%281-tan%5E2%28alpha%2F2%29%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, tan%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281-tan%5E2%28alpha%2F2%29%29.
The lessons Trigonometric functions of half argument and
                  Trigonometric functions of half argument - Examples









Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems
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