Question

Show \(\cos 3 A=4 \cos ^{3} A-3 \cos A\)

Short answer

Question: Prove the trigonometric identity \(\cos 3A = 4\cos^3 A - 3\cos A\). Answer: By expressing \(\cos 3A\) as \(\cos(2A + A)\) and using double angle formulas, we transformed the expression step by step and found that \(\cos 3A = 4\cos^3 A - 3\cos A\), thus proving the identity.

Step-by-step solution

Express \(\cos 3A\) as \(\cos(2A + A)\)

Using the formula for the cosine of a sum of angles, we have: \[\cos 3A = \cos(2A + A) = \cos 2A \cos A - \sin 2A \sin A\]

Express \(\cos 2A\) and \(\sin 2A\) using double angle formulas

The double angle formulas are: \[\cos 2A = 2\cos^2 A - 1\] \[\sin 2A = 2\sin A \cos A\] Using these formulas, we can replace \(\cos 2A\) and \(\sin 2A\) from Step 1: \[\cos 3A = (2\cos^2 A - 1)\cos A - (2\sin A \cos A)(\sin A)\]

Substitute \(\sin^2 A = 1 - \cos^2 A\)

Now, we need to use the identity \(\sin^2 A = 1 - \cos^2 A\) to rewrite the term with \(\sin A\) in Step 2: \[\cos 3A = (2\cos^2 A - 1)\cos A - 2(1 - \cos^2 A)(\cos A)(\sin A)\]

Simplify the expression

We will now simplify the expression obtained in Step 3 by distributing and combining like terms: \begin{align*} \cos 3A &= 2\cos^3 A - \cos A - 2\cos A \sin A + 2\cos^3 A \sin A \\ &= 2\cos^3 A - \cos A - 2\cos A (1 - \cos^2 A) \\ &= 2\cos^3 A - \cos A - 2\cos A + 2\cos^3 A \\ &= 4\cos^3 A - 3\cos A \end{align*} Therefore, we have shown that \(\cos 3A = 4\cos^3 A - 3\cos A\), as required.

Key concepts

Cosine

Cosine is one of the fundamental trigonometric functions, often abbreviated as 'cos'. It relates the angle of a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse. In the context of circular functions, it represents the x-coordinate of a point on the unit circle.

The function is periodic with a period of \(2\pi\), meaning that \(\cos(\theta + 2\pi) = \cos(\theta)\). This property is widely used in trigonometric identities and transformations. It's important to understand that cosine, like other trigonometric functions, can be applied beyond acute angles, accommodating negative angles and those exceeding \(360^\circ\) or \(2\pi\) radians.

Cosine plays a crucial role in many formulas, including the sum and double angle formulas. These help break down more complex angles into more manageable computations.

Double Angle Formulas

Double angle formulas are a set of trigonometric identities that express functions of double angles (e.g., \(2A\)) in terms of single angles (e.g., \(A\)). In our exercise, we specifically use the double angle formula for cosine and sine:

• \(\cos 2A = 2\cos^2 A - 1\)
• \(\sin 2A = 2\sin A \cos A\)

These formulas are particularly helpful as they simplify trigonometric expressions and equations that involve double angles. By reducing expressions like \(\cos 3A\) into more basic forms, they allow steps to be carried out using simpler algebraic manipulations.

In the given problem, the double angle formula for cosine helps to express \(\cos 2A\) in terms of \(\cos A\), making it easier to find an expression for \(\cos 3A\). This reduces the complexity of working directly with the angle \(3A\) by converting it into combinations of less complex angles, enabling more straightforward algebraic handling.

Sum of Angles

The sum of angles formula in trigonometry allows one to calculate the cosine or sine of a sum of two angles, such as \(A + B\), in terms of the sine and cosine of the individual angles. This is known as the angle addition formula. For cosine, it is expressed as:

\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]

This formula is crucial in the solution of \(\cos 3A\) because it is used to express \(\cos(2A + A)\). Breaking \(3A\) into \(2A + A\) helps decompose the problem into obtainable parts, which can be individually calculated using known identities and formulas.

In the step-by-step solution, the sum of angles formula allowed for the substitution into components that could make use of the double angle formulas, ultimately leading to the expression of \(\cos 3A\) in terms of \(\cos A\) alone. Understanding sum of angles gives students a powerful tool to tackle complex trigonometric expressions involving angles that are combinations of other angles.