Problem 1 - Algebra 2 and Trigonometry

Question

Sasha said that \(\sin \theta+\cos \theta=2\) has no solution. Do you agree with Sasha? Explain why or why not.

Step-by-Step Solution

Answer

Yes, I agree with Sasha. The equation has no solution because \(\sin \theta\) and \(\cos \theta\) cannot both be 1 simultaneously.

Step 1: Understanding the Trigonometric Identity

The given equation is \( \sin \theta + \cos \theta = 2 \). We know that the sine and cosine functions range from \(-1\) to \(1\) for all real numbers \(\theta\). Thus, the maximum value that \(\sin \theta + \cos \theta\) can achieve is \(1 + 1 = 2\).

Step 2: Assessing the Possibility

For \(\sin \theta + \cos \theta\) to equal \(2\), both \(\sin \theta\) and \(\cos \theta\) must simultaneously equal \(1\). However, \(\sin \theta = 1\) implies \(\theta = \frac{\pi}{2} + 2k\pi\), and \(\cos \theta = 1\) implies \(\theta = 2m\pi\), where \(k, m\) are integers. These two conditions cannot be satisfied by the same \(\theta\).

Step 3: Conclusion Based on Analysis

Since \(\sin \theta = 1\) cannot occur at the same \(\theta\) as \(\cos \theta = 1\), the sum \(\sin \theta + \cos \theta = 2\) is not possible. Thus, the equation \(\sin \theta + \cos \theta = 2\) has no solution.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry that describes the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The function is periodic, meaning it repeats values in cycles. The values of the sine function range between

• -1 and 1.
• The sine of \( heta \) is maximum, equal to \( 1 \), when \( heta \) is \( \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.
• The sine of \( heta \) is minimum, equal to \( -1 \), when \( \theta \) is \( \frac{3\pi}{2} + 2k\pi \).

This periodic nature is identified by sine's wave-like graph, called a sine wave. Each complete cycle, or period, of this wave spans an interval of \( 2\pi \). The graph crosses zero at integer multiples of \( \pi \). The sine function's continual oscillation between \( -1 \) and \( 1 \) is what makes it impossible for both sine and cosine of the same angle to each achieve \( 1 \), which in turn affects the prospect of solutions to equations like \( \sin \theta + \cos \theta = 2 \).

Cosine Function

Cosine, like the sine function, is a key trigonometric function that measures the ratio of the adjacent side to the hypotenuse of a right triangle. Cosine is also periodic and spans the same range as sine: it fluctuates between

• -1 and 1.
• The cosine of \( \theta \) reaches its maximum at \( 1 \) when \( \theta \) is \( 2m\pi \) where \( m \) is an integer.
• It achieves its minimum value of \( -1 \) at angles \( \pi + 2m\pi \).

The cosine function's oscillation further means that a cosine and sine value can't both be maximum at the same time, suggesting that the equation \( \sin \theta + \cos \theta = 2 \) can't have a valid real solution. Just like the sine wave, the cosine wave has a graceful arc that repeats every \( 2\pi \) units. Hence, understanding cosine's properties and behavior helps in analyzing equations involving trigonometric identities.

Trigonometric Functions Range

The range of trigonometric functions is crucial in understanding what values are achievable by sine, cosine, and other related functions. The primary trigonometric functions—sine and cosine—both range from

• -1 to 1.
• This defined range limits the possible outcomes of their sums and differences.

For any angle \( \theta \), the values of \( \sin \theta \) and \( \cos \theta \) are strictly bound within this interval. This makes it impossible for their sum to exceed \( 2 \), as it would require both functions to be at their maximum values simultaneously. Therefore, for \( \sin \theta + \cos \theta = 2 \), it isn't feasible due to the exclusive conditions under which both \( \sin \theta \) and \( \cos \theta \) reach their maximum value. Thus, understanding these range limits is vital when solving any problems related to trigonometric identities and ensures the consistency of solutions with realistic, mathematically verifiable outcomes.