# 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=, 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') = .

Problem 2
Prove that , if and are acute angles and , .

Solution
Calculate using the addition formula for tangents (see the lesson Addition and subtraction formulas in this module):

.

Since the angles and are acute and , we have
.

Problem 3
Prove that , if , and are acute angles and , and .

Solution
First, calculate using the addition formula for tangents (see the lesson Addition and subtraction formulas in this module):

.

Now, calculate using the same addition formula for tangents:

.

Since the angles and are acute and is positive (equal to ), the angle is acute.
Since the angles and are acute and is positive (equal to ), the angle is acute.
Since the angle is acute and , we have
.

Problem 4
If , show that
.

Solution
First, transform the sum as follows:
,

Next, represent as
.

Now, calculate
=

= =

= .

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.

 , , , , , . 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
 , , , . The lessons Powers of Trigonometric functions and                                                             Powers of Trigonometric functions - Examples

Trigonometric functions of multiply argument
 , , , . The lessons Trigonometric functions of multiply argument and                                                                 Trigonometric functions of multiply argument - Examples

Trigonometric functions of half argument
 , , , , , . 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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