Lesson Miscellaneous Trigonometry problems

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Miscellaneous Trigonometry problems


Problem 1
Find sin(22°30'), sin(22°30'), tan(22°30').

Solution
First, calculate sin(22°30').
Since 22°30' = 45°/2=pi%2F8, you can apply the formula of half argument for sines (see the lesson Trigonometric functions of half argument in this module):
sin(22°30') = .

Similarly, cos(22°30') = .

Hence,
tan(22°30') = sin(22°30')/cos(22°30') = sqrt%28%282-sqrt%282%29%29%2F%282%2Bsqrt%282%29%29%29.

Problem 2
Prove that alpha+%2B+beta+=+pi%2F4, if alpha and beta are acute angles and tan%28alpha%29+=+1%2F2, tan%28beta%29+=+1%2F3.

Solution
Calculate tan%28alpha%2Bbeta%29 using the addition formula for tangents (see the lesson Addition and subtraction formulas in this module):

.

Since the angles alpha and beta are acute and tan%28alpha%2Bbeta%29+=+1, we have
alpha+%2B+beta+=+pi%2F4.

Problem 3
Prove that alpha+%2B+beta+%2B+gamma+=+pi%2F4, if alpha, beta and gamma are acute angles and tan%28alpha%29+=+1%2F2, tan%28beta%29+=+1%2F5 and tan%28gamma%29+=+1%2F8.

Solution
First, calculate tan%28alpha%2Bbeta%29 using the addition formula for tangents (see the lesson Addition and subtraction formulas in this module):

.

Now, calculate tan%28alpha%2Bbeta%2Bgamma%29 using the same addition formula for tangents:

.

Since the angles alpha and beta are acute and tan%28alpha%2Bbeta%29 is positive (equal to 7%2F9), the angle alpha%2Bbeta is acute.
Since the angles alpha%2Bbeta and gamma are acute and tan%28alpha%2Bbeta%2Bgamma%29 is positive (equal to 1), the angle alpha%2Bbeta%2Bgamma is acute.
Since the angle alpha%2Bbeta%2Bgamma is acute and tan%28alpha%2Bbeta%2Bgamma%29=1, we have
alpha+%2B+beta+%2B+gamma+=+pi%2F4.

Problem 4
If alpha%2Bbeta%2Bgamma+=+pi, show that
.

Solution
First, transform the sum sin%282alpha%29+%2B+sin%282beta%29 as follows:
,

Next, represent sin%282gamma%29 as
sin%282gamma%29+=+2%2Asin%28gamma%29%2Acos%28gamma%29.

Now, calculate sin%282alpha%29+%2B+sin%282beta%29+%2B+sin%282gamma%29
=

                                    = =

                                    = .

You got what you were going to prove.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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








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 (this lesson)
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