Problem 88 - Algebra and Trigonometry

Question

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \cos \left(\frac{3 \pi}{2}-x\right)=-\sin x $$

Step-by-Step Solution

Answer

The equation is not an identity. An example of a value for \(x\) for which both sides are defined but not equal is \( x=\frac{\pi}{2} \)

Step 1 - Simplify the left side of the equation

We know that \( \cos (90 - α) = \sin α \) so we can simplify \( \cos \left(\frac{3 \pi}{2}-x\right) \) to \( \sin x\). Our equation thus becomes \( \sin x = -\sin x \).

Step 2 - Solve the simplified equation

The only way for \( \sin x = -\sin x \) to be true is if both sides equal zero. This is true for \(x = 0, x = \pi, x = 2\pi...\) and generally for \(x = n\pi\) where \(n\) is an integer.

Step 3 - Graph both sides of the equation

Even though our analysis shows that the equation is an identity for \(x = n\pi\), we should verify this visually. Graph \( \sin x \) and \( -\sin x \). The two graphs coincide only on the x-axis, which supports our analytical finding.

Step 4 - Determine Value of \(x\)

Since the two sides coincide on the x-axis, we can say that the equation is not an identity. A value of \(x\) for which both sides are defined but not equal is, for example, \( x=\frac{\pi}{2} \) because \(\sin \frac{\pi}{2} \neq -\sin \frac{\pi}{2} \)

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential for solving many types of mathematical problems, especially those involving periodic phenomena. Trigonometric functions include sine (\textnormal{sin}), cosine (\textnormal{cos}), and tangent (\textnormal{tan}), along with their reciprocals cosecant (\textnormal{csc}), secant (\textnormal{sec}), and cotangent (\textnormal{cot}).

At the heart of these functions are the relationships they represent between the angles and sides of a right triangle. However, these functions extend beyond right triangles, applying to any angles and even to points on a unit circle centered at the origin of a coordinate system. Here, the (\textnormal{sin}) of an angle is the y-coordinate, the (\textnormal{cos}) is the x-coordinate, and the (\textnormal{tan}) is the ratio of y over x.

The functions have a periodic nature, with \textnormal{sin} and \textnormal{cos} repeating every \(2\textnormal{pi}\) radians and \textnormal{tan} every \(\textnormal{pi}\) radians. In solving equations, this periodicity means that there are infinitely many solutions, often taking the form of \(x = n\textnormal{pi} + \textnormal{something}\), where \(n\) is an integer. Understanding this concept is critical when graphing these functions or applying them to real-world problems.

Graphing Trigonometric Functions

Graphing trigonometric functions helps students visualize the periodic nature and properties of these functions. When graphing, it's critical to consider the amplitude, period, phase shift, and vertical shift.

• Amplitude refers to the maximum displacement from the horizontal axis.
• Period is the distance along the x-axis for one complete cycle of the function.
• Phase shift moves the graph left or right from its usual start position.
• Vertical shift moves the entire graph up or down.

Using a consistent scale and marking key points, such as the intercepts and maximum and minimum points, are essential steps. For example, the basic \textnormal{sin} and \textnormal{cos} functions have a period of \(2\textnormal{pi}\), so marking intervals of \(\frac{\textnormal{pi}}{2}\) on the x-axis can provide clarity.

By examining where functions like \(\textnormal{sin } x\) and \(-\textnormal{sin } x\) intersect, we can confirm the solutions to equations or examine their identities. In the case of the exercise given, plotting these two functions on a graph can demonstrate visually that although they intersect at certain points, they are not identical functions, since their values differ for most values of \(x\). This graphing process reinforces understanding of the functions' behavior and the verification of trigonometric identities.

Solving Trigonometric Equations

Solving trigonometric equations can range from straightforward to quite complex, depending on the forms and transformations involved. The key is to use trigonometric identities to simplify the equation to a basic form which can be easily solved.

Some common strategies include:

• Using Pythagorean identities to transform one function into another
• Applying angle addition or subtraction formulas to combine or break apart terms
• Factoring to find common solutions
• Using inverse trigonometric functions to find specific angle measures

In the exercise provided, a trigonometric equation was simplified using an angle subtraction identity. Once simplified, the solution showed when the two sides could be equal, which led to the conclusion that the original equation was not an identity for all values of \(x\), but had specific solutions. This method—relying not only on the algebraic process but also on the graphical comparison—provides a holistic understanding of how the equation behaves for all real numbers and enhances the ability to solve more complex trigonometric equations.