Problem 44 - Trigonometry

Question

Find all radian solutions using exact values only. $$\sin x-\cos x=1$$

Step-by-Step Solution

Answer

There is no solution for \( \sin x - \cos x = 1 \) in real numbers.

Step 1: Identify the Trigonometric Equation

The equation given is \( \sin x - \cos x = 1 \). Our goal is to find values of \( x \) such that this equation holds true.

Step 2: Isolate One Trigonometric Function

To isolate one of the functions, we can rearrange the equation: either \( \sin x = \cos x + 1 \) or \( \cos x = \sin x - 1 \). Let's use \( \sin x = \cos x + 1 \).

Step 3: Compare Maximum Values

The maximum value of both \( \sin x \) and \( \cos x \) is 1, but \( \cos x + 1 \) can only equal 2, so \( \sin x = 2 \) must hold, which is impossible since \( \sin x \leq 1 \). Hence, there's no possible solution.

Step 4: Verify with Known Identities

Since mathematical functions cannot exceed their inherent limits (\( \sin x \leq 1 \) and \( \cos x \leq 1 \)), and considering \( \cos x + 1 \) results exceeding one, there are no angles \( x \) that satisfy the original equation when evaluated under real conditions.

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry, defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This function is abbreviated as \( \sin \). Here are some important points about the sine function:

• Range and Domain: The sine function can have values ranging from -1 to 1. Its domain includes all real numbers, meaning it can accept any angle measured in radians or degrees.
• Behavior and Symmetry: The sine function is periodic with a period of \(2\pi\). It is an odd function, which means that \(\sin(-x) = -\sin(x)\), showing its symmetry about the origin.
• Graphical Representation: The graph of \(\sin x\) is a smooth, wave-like curve that repeats every \(2\pi\) units. It crosses the x-axis at multiples of \(\pi\).

Understanding the sine function's limits and behavior is crucial when solving equations like \( \sin x - \cos x = 1 \). In our exercise, the theoretical calculation led us to an impossible scenario, requesting \( \sin x \) to exceed its maximum possible value (which is 1). Thus, recognizing these fundamental aspects assures us that no solution exists under these conditions.

Cosine Function

The cosine function, much like the sine function, is quintessential in trigonometry. It is the ratio of the length of the adjacent side to the hypotenuse in a right-angle triangle, commonly denoted as \(\cos\). Key points to know about the cosine function include:

• Range and Domain: Similar to the sine function, \(\cos x\) also ranges between -1 and 1. Any real number can be input into the cosine function, making its domain all real numbers.
• Symmetry and Behavior: The cosine function is periodic with a period of \(2\pi\). It is an even function, implying that \(\cos(-x) = \cos(x)\), indicating symmetry about the y-axis.
• Graphical Features: The graph of \(\cos x\) is also a smooth curve, wave-like, and repeats every \(2\pi\) units. However, it starts at 1 when \(x = 0\), in contrast to the sine curve.

In the context of our problem, adjusting \(\cos x\) by adding 1 led to a theoretical maximum of 2, which actually violates \(\cos x\)'s inherent range. Thus this impossibility further highlights why there is no solution to \(\sin x - \cos x = 1\) with real numbers.

Radian Solutions

Radian solutions play a key role in trigonometry because they offer a natural way to express angles. One radian is the angle created when the arc length is equal to the radius of the circle.

• Definition: Radians are a measure for angles. While degrees divide a circle into 360 parts, radians divide it based on the circle's radius, encompassing \(2\pi\) for a full revolution.
• Conversion: The conversion between degrees and radians is critical: \(180^\circ = \pi\) radians. This conversion helps us translate traditional degree solutions into radians when solving trigonometric equations.
• Advantages: Radians simplify mathematical expressions, especially when working with calculus or complex trigonometric functions. They also make periodic functions harmonious without needing conversion factors.

In our original exercise, expecting to solve \(\sin x - \cos x = 1\) in radians reinforces the need to understand the limits within real scenarios. It solidifies why no radian solutions exist since the theoretical calculations exceed possible values of \(\sin\) and \(\cos\). Recognizing the compatibility of radian measures with trigonometric limits is essential for solving these equations.