Lesson Addition and subtraction formulas - Examples

Addition and subtraction formulas - Examples

The addition and subtraction Trigonometry formulas are:











The proofs of these formulas are presented in the lesson Addition and subtraction formulas in this module.
Below are examples of application of these formulas.

Example 1
Find cos(75°), sin(75°) and tan(75°).

Solution
Note that 75° = 30° + 45°.
Calculate cos(75°) by applying the addition formula:

cos(75°) = cos(30°)*cos(45°) - sin(30°)*sin(45°) = .

Calculate sin(75°) by applying the addition formula:

sin(75°) = sin(30°)*cos(45°) + cos(30°)*sin(45°) = .

Now, calculate tan(75°) as the fraction sin(75°)/cos(75°):

tan(75°) = sin(75°)/cos(75°) = , as it follows from the lines above.
Simplify this:

.

Or, you can calculate tan(75°) by applying the addition formula for tangents:
tan(75°) = (tan(30°) + tan(45°))/(1 - tan(30°)*tan(45°)) = .
Simplify this:

.

Thus, you see that both calculations produce the same result for tan(75°), namely, .

Example 2
Find cos(15°), sin(15°) and tan(15°).

Solution
Note that 15° = 45° - 30°.
Calculate cos(15°) by applying the subtraction formula:

cos(15°) = cos(45°)*cos(30°) + sin(45°)*sin(30°) = .

Calculate sin(15°) by applying the subtraction formula:

sin(15°) = sin(45°)*cos(30°) - cos(45°)*sin(30°) = .

Now, calculate tan(15°) as the fraction sin(15°)/cos(15°):

tan(15°) = sin(15°)/cos(15°) = , as it follows from the lines above.
Simplify this:

.

Or, you can calculate tan(15°) by applying the subtraction formula for tangents:
tan(15°) = (tan(45°) - tan(30°))/(1 + tan(45°)*tan(30°)) = .
Simplify this:

.

Thus, you see that both calculations produce the same result for tan(15°), namely, .

Another way to solve the Example 2 is to note that 15° = 90° - 75° and then to apply the formulas for the complementary angle
and to use results of the Example 1:

cos(15°) = sin(75°) = ,

sin(15°) = cos(75°) = ,

tan(15°) = cot(75°) = 1/tan(75°) = .

Example 3
Find , and , if , and and are the first quadrant angles.

Solution
Since and is the first quadrant angle, we have
.

Since and is the first quadrant angle, we have
.

Now apply the addition formulas:

,

.

, as it follows from the previous two lines, OR

from the addition formula for tangents.

Both calculated results for tangents are identical.



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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

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The lessons Addition and subtraction formulas and                      Addition and subtraction formulas - Examples (this lesson)

Addition and subtraction of trigonometric functions

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

Product of trigonometric functions

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

Powers of trigonometric functions

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

Trigonometric functions of multiply argument

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

Trigonometric functions of half argument

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The lessons Trigonometric functions of half argument and                   Trigonometric functions of half argument - Examples

Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems