The lessons Addition and subtraction formulas and
Addition and subtraction formulas - Examples
The formulas for the Trigonometric functions of multiply argument are:
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 = 15°. Note that = 30° and use the formula for cosines of the double argument:
Substitute = 15°, = 30° and = cos(30°) = into this formula. You get the equation
= 2*cos^2(15°) - 1, or
Since you already calculated cos(15°), you can easily find sin(15°):
sin^2(15°) = 1 - cos^2(15°) = ,
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 = 18°.
Then = 90°,
hence = 90°-.
Therefore,
= ,
and consequently
(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
.
Since is not equal to zero, you can divide both sides of the precedent equality by . You get the equation
.
Now, introduce for short and replace in the precedent formula. You get the equation
,
or, after simplifying,
.
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°) = .
Since you already calculated sin(18°), you can easily find cos(18°):
cos^2(18°) = 1 - sin^2(18°) = .
Hence,
cos(18°) = .
Consequently,
tan(18°) = sin(18°)/cos(18°) = .
Example 3
Find sin(36°), cos(36°) and tan(36°).
You just learned (from the precedent Example ) that cos(18°) = .
Apply the formula of cosines for the double argument to calculate cos(36°):
cos(36°) = cos(2*18°) = 2*cos^2(18°) - 1 = .
Now, you can easily calculate sin(36°):
sin^2(36°) = 1 - cos^2(36°) = ,
therefore sin(36°) = .
Consequently
tan(36°) = sin(36°)/cos(36°) = .
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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
, .