91Ó°ÊÓ

In trigonometry, the cosine identity is a key tool used to solve equations involving cosine functions. In essence, the identity states that if two cosine values are equal, i.e., \( \cos(A) = \cos(B) \), then their corresponding angles are related by:

• \( A = 2k\pi \pm B \)

where \( k \) is any integer. This relationship arises because the cosine function is periodic and repeats every \( 2\pi \) radians. Whenever you encounter a problem where you need to find angles with equal cosines, you can use this identity to express the possible solutions.
Applying this principle allows you to systematically find all angles for which \( \cos(4x) = \cos(2x) \). By substituting different integer values for \( k \), you can derive the specific angles or values of \( x \) that satisfy the equation within a specified interval.

Finding solutions to trigonometric equations involves understanding their periodic nature and how they behave within a defined interval. The exercise requires finding all angles \( x \) such that \( \cos(4x) = \cos(2x) \) within the interval \([0, 2\pi)\). To start, you break down the equation using the cosine identity, leading to two potential equations:

• \( 4x = 2k\pi + 2x \)
• \( 4x = 2k\pi - 2x \)

Each equation gives a set of solutions when solved for \( x \):

• From \( 4x = 2k\pi + 2x \), we get \( x = k\pi \).
• From \( 4x = 2k\pi - 2x \), we get \( x = \frac{k\pi}{3} \).

Within the interval \([0, 2\pi)\), the possible integer values for \( k \) help determine the specific solutions. This problem illustrates the method of systematically exploring these values to find all \( x \) within the desired range that satisfy the initial condition.

Radian measure is a way of expressing the size of angles based on the radius of a circle. Unlike degrees, radians relate directly to the dimensions of a circle, making them very useful in calculus and trigonometry. A full circle corresponds to \( 2\pi \) radians. Therefore, an angle of \( 2\pi \) radians means completing a full circle.

• \( \pi \) radians is half of a circle, equivalent to 180 degrees.
• A radian itself is the angle created when the arc length is equal to the radius of the circle.

In the exercise, finding solutions in the interval \([0, 2\pi)\) requires calculating various angular measures in radians. Converting between radians and degrees can often simplify understanding but working directly with radians is essential for accurately solving trigonometric equations. Using radians simplifies the calculations involved with periodic functions like sine and cosine, especially when seeking all solutions in a given interval.