Trigonometric functions of multiply argument - 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.
Find cos(15°), sin(15°), tan(15°).
First, find cos(15°).
= 15°. Note that
= 30° and use the formula for cosines of the double argument:
= 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°) =
tan(15°) = sin(15°)/cos(15°) =
Note that sin(15°), cos(15°) and tan(15°) were just calculated differently in the lessons
Addition and subtraction formulas - Examples
Product of trigonometric functions - Examples
in this module.
Please make sure that all relevant results from these lessons are identical.
Find sin(18°), cos(18°) and tan(18°).
Let us denote
(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
is not equal to zero, you can divide both sides of the precedent equality by
. You get the equation
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
Only the first root fits (the second root doesn't fit due to its sign).
So, the answer
Since you already calculated sin(18°), you can easily find cos(18°):
cos^2(18°) = 1 - sin^2(18°) =
= sin(18°)/cos(18°) =
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(2*18°) = 2*cos^2(18°) - 1 =
Now, you can easily calculate sin(36°):
sin^2(36°) = 1 - cos^2(36°) =
= sin(36°)/cos(36°) =
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
Addition and subtraction of trigonometric functions
Product of trigonometric functions
Powers of trigonometric functions
Trigonometric functions of multiply argument
Trigonometric functions of half argument
Miscellaneous Trigonometry problems
The lesson Miscellaneous Trigonometry problems
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