Problem 18 - Calculus

Question

Find all angles \(s\) and \(t\) between 0 and \(2 \pi\) where \(\sin (s+t)=\) \(\sin s+\sin t\).

Step-by-Step Solution

Answer

Solutions are \((s, t) = (0, 0), (\pi, 0), (0, \pi), (\pi, \pi)\).

Step 1: Use Sine Addition Formula

The sine addition formula is \[sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\]In our problem, apply this to \(s + t\):\[sin(s + t) = sin(s)cos(t) + cos(s)sin(t)\]

Step 2: Set the Equation Equal

Substitute the sine addition formula expression into the given equation:\[sin(s)cos(t) + cos(s)sin(t) = sin(s) + sin(t)\]

Step 3: Rearrange the Equation

Rearrange the terms to bring all terms involving \(sin(s)\) and \(sin(t)\) on one side:\[sin(s)cos(t) - sin(s) = sin(t) - cos(s)sin(t)\]

Step 4: Factor Out Common Terms

Factor out \(sin(s)\) and \(sin(t)\) from their respective sides:\[sin(s)(cos(t) - 1) = sin(t)(1 - cos(s))\]

Step 5: Analyze Possible Cases

Since this equation involves terms \(sin(s)=0\), \(cos(t)=1\),\(sin(t)=0\), and \(cos(s)=1\), consider each scenario to solve the equation.

Step 6: Solving \(sin(s) = 0\) or \(sin(t) = 0\)

If \(sin(s) = 0\), then \(s = 0\) or \(s = \pi\). If \(sin(t) = 0\), then \(t = 0\) or \(t = \pi\).

Step 7: Solving \(cos(t) = 1\) or \(cos(s) = 1\)

If \(cos(t) = 1\), then \(t = 0\). If \(cos(s) = 1\), then \(s = 0\).

Step 8: Combine Solutions

Combining the solutions from these specific conditions, solutions for \((s,t)\) are: For \(s = 0\) or \(s = \pi\) and \(t = 0\) or \(t = \pi\), possible angles are \(s = 0, t = 0; s = \pi, t = 0; s = 0, t = \pi;\) and \(s = \pi, t = \pi\).

Key Concepts

Sine Addition Formula

The Sine Addition Formula is a fundamental tool in trigonometry. Explained simply, it helps us to find the sine of an angle that results from adding two other angles. The formula is expressed as:\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]Here, \( a \) and \( b \) are individual angles, and the formula tells us how to express the sine of their sum using the sines and cosines of the individual angles.

• Imagine you are trying to determine the sine of the sum of two angles that you know; this formula allows you to calculate that easily.
• It breaks down one seemingly complex sine into familiar parts — separate sines and cosines.

In the context of our exercise, we used this formula to transform \( \sin(s + t) \) into \( \sin(s)\cos(t) + \cos(s)\sin(t) \). This transformation is crucial for setting and manipulating the equation to eventually isolate solutions.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are algebraic tools that enable us to transform and simplify trigonometric expressions.

• These identities are integral parts of solving equations involving trigonometric functions. They allow us to swap one form of a trigonometric expression for another, aiding in simplification.
• Familiar identities include the basic Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

In this problem, the key identity driving the solution is the manipulation using the Sine Addition Formula. This particular identity reveals the core technique of expressing the sum of two angles differently, simplifying the problem.With the equation manipulated, we factored it based on common trigonometric outputs like \( \sin \) and \( \cos \). Understanding that if \( \sin(s) \cos(t) + \cos(s) \sin(t) = \sin(s) + \sin(t) \), all trigonometric components must be reconciled aids in solving such equations.

Angle Solutions

Finding angle solutions in trigonometric equations revolves around identifying specific angle conditions that satisfy the equation within a given range.

• Solutions often rely heavily on the properties of trigonometric functions, such as their periodic nature.
• Solving involves considering special angles where sine and cosine take on simple values — typically 0, 1, or -1.

In the exercise, the task was to identify angles \( s \) and \( t \) within the interval \( [0, 2\pi] \) such that the equation \( \sin(s + t) = \sin(s) + \sin(t) \) holds true.By analyzing specific cases where \( \sin(s) = 0 \), \( \sin(t) = 0 \), \( \cos(t) = 1 \), and \( \cos(s) = 1 \), we derive the unique sets of angle pairs: - \( s = 0, t = 0 \)- \( s = \pi, t = 0 \)- \( s = 0, t = \pi \)- \( s = \pi, t = \pi \)These solutions reflect the angles where our equation neatly aligns within the stipulated range.