Problem 173
Question
$$ \cos 6 a=32 \cos ^{6} a-48 \cos ^{4} a+18 \cos ^{2} a-1 $$
Step-by-Step Solution
Verified Answer
The original equation holds true for any 'a'.
1Step 1: Recognize the Pattern and Simplify the Equation
The equation \(\cos 6 a = 32 \cos ^{6} a - 48 \cos^{4} a + 18 \cos^{2} a - 1\) has coefficients following the binomial theorem pattern. Therefore, the equation can be simplified to \(2\cos(a) = \cos(6a)\).
2Step 2: Apply the Multiple-Angle Formula
Next, apply the multiple-angle formula on the left side of the equation to rewrite the trigonometric function: \(2\cos(a) = 32\cos^5(a) - 80\cos^3(a) + 50\cos(a) - 1\). This simplifies to \(\cos(6a) = 32\cos^5(a) - 80\cos^3(a) + 50\cos(a) - 1\).
3Step 3: perform the comparison
We can now compare the function \(\cos(6a)\) and the equation \(32\cos^5(a) - 80\cos^3(a) + 50\cos(a) - 1\). If these two are equal for all a, that implies that the original equation holds true for any a. Therefore, the original equation holds true for any a.
Key Concepts
Understanding Trigonometric FunctionsThe Binomial Theorem in TrigonometryApplying the Cosine Multiple-Angle Formula
Understanding Trigonometric Functions
Trigonometric functions are the backbone of trigonometry and are essential in describing the relationships between the angles and sides of triangles. The most basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each defined as ratios of sides in a right-angled triangle. For an angle \(a\), cosine specifically relates to the adjacent side over the hypotenuse, written as \( \text{cos}(a) = \frac{adjacent}{hypotenuse}\).
These functions are also defined for any angle extending to the Cartesian plane which allows for a broader application in various fields including physics, engineering, and even finance. Understanding these functions and their properties is a prerequisite for delving into more complex topics like multiple-angle identities.
These functions are also defined for any angle extending to the Cartesian plane which allows for a broader application in various fields including physics, engineering, and even finance. Understanding these functions and their properties is a prerequisite for delving into more complex topics like multiple-angle identities.
The Binomial Theorem in Trigonometry
The binomial theorem provides a powerful way to expand expressions that are raised to a power. In trigonometry, this theorem can be used to expand trigonometric functions involving binomial expressions. For example, \((x + y)^n\) can be expanded using the binomial coefficients, which are the numbers in Pascal's triangle.
When dealing with powers of cosine such as \(\text{cos}^n(a)\), one can apply the binomial theorem to expand expressions involving terms like \(\text{cos}^6(a)\) or \(\text{cos}^4(a)\). These expansions are useful in deriving multiple-angle trigonometric identities, which show the cosine of a multiple of an angle in terms of powers of the cosine of the original angle. The pattern of coefficients in such identities often mirrors those from Pascal's triangle.
When dealing with powers of cosine such as \(\text{cos}^n(a)\), one can apply the binomial theorem to expand expressions involving terms like \(\text{cos}^6(a)\) or \(\text{cos}^4(a)\). These expansions are useful in deriving multiple-angle trigonometric identities, which show the cosine of a multiple of an angle in terms of powers of the cosine of the original angle. The pattern of coefficients in such identities often mirrors those from Pascal's triangle.
Applying the Cosine Multiple-Angle Formula
The cosine multiple-angle formula is a fundamental identity in trigonometry that expresses the cosine of a multiple of an angle in terms of powers of the cosine of a single angle. For instance, the formula for \( \text{cos}(2a) \) is \( \text{cos}^2(a) - \text{sin}^2(a) \) or, using the Pythagorean identity \( \text{sin}^2(a) + \text{cos}^2(a) = 1 \), as \( 2\text{cos}^2(a) - 1 \).
This multiple-angle identity can be extended to higher multiples such as \( \text{cos}(3a) \), \( \text{cos}(4a) \), and beyond. In our problem, we are dealing with \( \text{cos}(6a) \), which can be expressed as a polynomial in \( \text{cos}(a) \) using multiple-angle formulas derived from combinations of lower-order formulas. This is vital in solving complex trigonometric equations where the angles are multiples of a known angle.
This multiple-angle identity can be extended to higher multiples such as \( \text{cos}(3a) \), \( \text{cos}(4a) \), and beyond. In our problem, we are dealing with \( \text{cos}(6a) \), which can be expressed as a polynomial in \( \text{cos}(a) \) using multiple-angle formulas derived from combinations of lower-order formulas. This is vital in solving complex trigonometric equations where the angles are multiples of a known angle.
Other exercises in this chapter
Problem 171
$$ \cos a \cos \left(60^{\circ}-a\right) \cos \left(60^{\circ}+a\right)=\frac{1}{4} \cos 3 a $$
View solution Problem 172
$$ \cot a+\cot \left(60^{\circ}+a\right)-\cot \left(60^{\circ}-a\right)=3 \cot 3 a $$
View solution Problem 174
$$ \cos ^{3} \theta+\cos ^{3}\left(120^{\circ}+\theta\right)+\cos ^{3}\left(240^{\circ}+\theta\right)=\frac{3}{4} \cos 3 \theta $$
View solution Problem 175
$$ \text { If } \cos \theta=\frac{1}{2}\left(a+\frac{1}{a}\right), \text { then prove that } \cos 3 \theta=\frac{1}{2}\left(a^{3}+\frac{1}{a^{3}}\right) \text {
View solution