Problem 60 - Algebra and Trigonometry

Question

Graph \(f\) and \(g\) on the same axes, and find their points of intersection. $$f(x)=\sin x-1, g(x)=\cos x$$

Step-by-Step Solution

Answer

The points of intersection can be estimated graphically or solved approximately, as \(\sin x - \cos x = 1\) is complex analytically.

Step 1: Graph the Functions

Begin by graphing the two functions, \(f(x) = \sin x - 1\) and \(g(x) = \cos x\), on the same set of axes. For \(f(x)\), shift the sine wave down by 1 unit. For \(g(x)\), graph the standard cosine wave. Use a range of \(x\) values such as \([0, 2\pi]\) or \([-2\pi, 2\pi]\) to observe the periodic behavior.

Step 2: Identify Points of Intersection Graphically

Look at the graph of the two functions and identify where they intersect. These points occur where the y-values of \(f(x)\) and \(g(x)\) are equal, indicating they are at the same point on the graph.

Step 3: Set the Equations Equal

To find the exact points of intersection, set \(f(x)\) equal to \(g(x)\). This means solving the equation \(\sin x - 1 = \cos x\).

Step 4: Solve the Equation \(\sin x - 1 = \cos x\)

Rearrange the equation to \(\sin x - \cos x = 1\). One method to solve this is using trigonometric identities or substitution for \(x\). Alternatively, graphically approximate solutions within known ranges or use numerical methods for precision.

Step 5: Verify Solutions

Substitute the found values of \(x\) back into both functions to verify that they yield the same y-value, confirming they intersect at these points. However, as it may be non-trivial to solve exactly, approximate solutions and verify graphically or with a calculator.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in understanding periodic behavior. They are functions of an angle, often encountered in the context of unit circles.
The sine function, denoted as \(\sin x\), represents the y-coordinate of a point on the unit circle as the angle \(x\) varies. Its graph is a smooth, repetitive wave that oscillates between -1 and 1. The function \(f(x) = \sin x - 1\) indicates that this wave is shifted downwards by 1 unit.
The cosine function, \(\cos x\), represents the x-coordinate of the same point. Its graph is also a wave that oscillates between -1 and 1, but it starts at 1 when \(x = 0\). These functions are periodic with a period of \(2\pi\), meaning they repeat their pattern every \(2\pi\) units along the x-axis.

Points of Intersection

The points of intersection between two functions occur where their y-values are equal at the same x. For the functions \(f(x) = \sin x - 1\) and \(g(x) = \cos x\), intersections happen where \(\sin x - 1 = \cos x\).
Identifying these points is crucial as they provide solutions where both functions coincide. This can be visualized graphically by observing where the graphs overlap.
When the graphs intersect, it implies that the two functions have the same output or y-value for certain x-values, which is key to understanding their relationship and solving related equations.

Solving Equations

Solving the equation \(\sin x - 1 = \cos x\) is a way to find exact points of intersection. To solve, one option is to bring the terms involving trigonometric functions together and isolate them into manageable forms:

• Rearrange to \(\sin x - \cos x = 1\).
• Use trigonometric identities (e.g., \(\sin x = \sqrt{1-\cos^2 x}\) or angles formulas).
• Implement substitution methods if necessary.

This equation needs careful handling due to the non-linear nature of sine and cosine functions. Graphical methods or numerical solutions can also approximate solutions if the algebra becomes complex. Always verify solutions by checking they satisfy the original equation.

Graphical Solutions

Graphical solutions provide a visual approach to solving equations, invaluable when equations are tricky or resistant to algebraic manipulation. By plotting \(f(x) = \sin x - 1\) and \(g(x) = \cos x\):

• The functions' graphs reveal intersections visually.
• You can estimate the intersection points' coordinates directly from the graph.
• Using graphing calculators or software can increase precision.

Graphing both functions provides insights into their behavior and helps in understanding how they interact. It's a powerful tool in analyzing and resolving equations involving trigonometric functions as supplementary to algebraic solutions.