Solve each equation for x, where x is restricted to the given interval.
y = ― 2 cos 5x , for x in [0, π/5]

Verified step by step guidance

Step 1: Start by isolating the cosine function. Since the equation is y = -2 cos(5x), divide both sides by -2 to get cos(5x) = -y/2.

Step 2: Use the inverse cosine function to solve for 5x. This gives 5x = cos^(-1)(-y/2).

Step 3: Solve for x by dividing both sides of the equation by 5, resulting in x = (1/5) * cos^(-1)(-y/2).

Step 4: Consider the range of the cosine function and the given interval for x, [0, π/5]. The cosine function is periodic, so ensure that the solution for x falls within this interval.

Step 5: If necessary, use the properties of the cosine function to find additional solutions within the interval by considering the symmetry and periodicity of the cosine function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π, meaning it repeats its values every 2π units. Understanding the behavior of the cosine function, including its maximum and minimum values, is essential for solving equations involving cosine.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to find angles when the value of a trigonometric function is known. For example, if cos(θ) = y, then θ = arccos(y). These functions are crucial for solving equations where the variable is inside a trigonometric function, allowing us to isolate the angle and find its value.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [0, π/5] indicates that x can take any value from 0 to π/5, inclusive. Understanding how to interpret and work within specified intervals is important for determining valid solutions to trigonometric equations.

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