Linearize the Trigonometric Powers Part 2

Geometry Level 2

$\cos^3 \theta = a_3 \cos (3 \theta) + a_1 \cos( \theta )$

Above shows a trigonometric identity for constants $a_1, a_3$ .

What is the value of $a_1 \div a_3$ ?

See Part 1 , Part 3 , and Part 4 .

The answer is 3.

1 solution

Jake Lai
May 6, 2015

From the compound angle formula and double-angle formula, we may obtain the identity $4\cos^{3} \theta-3\cos \theta = \cos 3\theta$ . Rearrangment gives

$\cos^{3} \theta = \frac{1}{4} \cos 3\theta+\frac{3}{4} \cos \theta$

As such, $a_{1} = 3a_{3} = \frac{3}{4}$ , and $a_{1} \div a_{3} = \boxed{3}$ .

Linearize the Trigonometric Powers Part 2

The answer is 3.

1 solution

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