Lesson Miscellaneous Trigonometry problems

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Miscellaneous Trigonometry problems


Problem 1

Find  sin(22°30'),  cos(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 = ( simplify ) = sqrt%28+%282-sqrt%282%29%29%5E2%2F%28+%282%2Bsqrt%282%29%29%2A%282-sqrt%282%29%29+%29+%29 = %282+-sqrt%282%29%29%2Fsqrt%284-2%29+%29 = %282-sqrt%282%29%29%2Fsqrt%282%29+%29 = sqrt%282%29-1.

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.



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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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