Problem 77 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\sin x+\cos x=1$$

Step-by-Step Solution

Answer

The solutions to the equation \( \sin x+\cos x=1 \) within the interval [0, 2π) are: x = 0, π/2, π, 3π/2.

Step 1: Use Pythagorean identity

Start by observing that the equation could be transformed using the Pythagorean identity. Square both sides of the equation to get: \[(\sin x+\cos x)^2=1^2 \]which simplifies to \[\sin^2 x+ 2 \sin x \cos x +\cos^2 x = 1\]

Step 2: Apply sin^2(x) + cos^2(x) = 1

Use the Pythagorean identity that states \(\sin^2(x) + \cos^2(x) = 1\). This simplify the equation to: \[ 1 + 2\sin x \cos x = 1\]which reduces to \[2\sin x \cos x = 0\]

Step 3: Apply Double-Angle Identity

We can then apply the double-angle identity \( \sin(2x) = 2\sin(x)\cos(x) \) to further simplify the equation, resulting in\[ \sin(2x) = 0\]

Step 4: Find solutions within the interval [0, 2π]

Solutions to the equation \( \sin(2x) = 0 \) lie at integer multiples of π. However, remember that we're solving for 'x', not '2x'. Therefore, 'x' lies at half the solutions we found. That's to say, \( x = \frac{n\pi}{2} \), where 'n' is an integer. To limit these solutions to the provided interval [0, 2π], we have: \[ x = 0, \: \frac{\pi}{2}, \: \pi, \: \frac{3\pi}{2} \]

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial for solving various trigonometric equations. This identity is an extension of the famous Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In trigonometric form, the Pythagorean identity is expressed as:
\[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]
This identity holds true for any angle x and is a fundamental relationship between the sine and cosine functions.When you come across a trigonometric equation that involves both sin and cos, it's often helpful to check if you can apply this identity. By rewriting one or both of these functions in terms of the other using the Pythagorean identity, you can simplify the equation and make it easier to solve.

Double-Angle Identity

The double-angle identities in trigonometry are a set of equations that relate the sine, cosine, and tangent of an angle to the sine, cosine, and tangent of its double. One common double-angle identity for sine is:
\[ \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \]
This identity can be extremely valuable when solving trigonometric equations that feature products of sine and cosine functions. As seen in the provided solution, transforming an equation with a sine and cosine product into a double-angle form can dramatically simplify the problem.
For instance, if an equation arrives at a point where it shows a product of sin(x) and cos(x), employing the double-angle identity can reduce it to a single trigonometric function of a different angle, which is often easier to solve or evaluate within a given interval.

Sin and Cos Functions

The sine (sin) and cosine (cos) functions are two of the most important trigonometric functions. They are used to describe the relationships between the angles and lengths of triangles, particularly right triangles, as well as to model periodic phenomena such as sound and light waves.

Properties of Sin and Cos Functions

• The sin function takes an angle as input and gives the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
• The cos function, similarly, gives the ratio of the length of the adjacent side to the length of the hypotenuse.
• Both sin and cos functions are periodic with a period of \(2\text{π}\), meaning they repeat their values in regular intervals of \(2\text{π}\).
• The maximum value of both functions is 1, and the minimum is -1, corresponding to the points where the angle leads to a point on the unit circle at its farthest from the origin, whether above, below, or on either side.

The values taken by these functions at specific angles are often critical to solving trigonometric equations. Being familiar with their fundamental properties, such as their periodicity, or the specific values they take at important angles like \(0, \frac{\text{π}}{2}, \text{π}, \text{and} \frac{3\text{π}}{2}\), can simplify the process of finding solutions to these equations within the desired intervals.