Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta$$

Step-by-Step Solution

Answer

The identity \(\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta\) is valid since all the algebraic manipulations support this.

Step 1: Understand the given identity

The given identity to be verified is \(\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta\)

Step 2: Simplify the left side

The \(\csc \theta\) is equivalent to \(1 / \sin \theta\). Therefore, \(\sin \theta \csc \theta\) simplifies to \( \sin \theta * (1/\sin \theta) = 1\). So, the left side of the equation becomes \(1 - \sin^{2}\theta\).

Step 3: Recognize the Pythagorean Identity

From trigonometry, we know that \(\sin^{2}\theta + \cos^{2}\theta = 1\). We can then rearrange this identity to \(\cos^{2}\theta = 1 - \sin^{2}\theta.\)

Step 4: Compare the results

After simplifying the left side of the equation in step 2, it is evident that the result is similar to the right side of the equation as rearranged in step 3.

Step 5: Check the result graphically

Next, you can use a graphing utility to plot the function \(y = \sin \theta \csc \theta-\sin ^{2} \theta\) and \(y = \cos^2\theta\). If the identity holds true, both graphs will be the same.

Key Concepts

Trigonometric Pythagorean Identity

The trigonometric Pythagorean identity is one of the most fundamental relationships in trigonometry. It relates the squares of the sine and cosine functions of any angle \theta. According to this identity:
\[\sin^2\theta + \cos^2\theta = 1.\]This means that, for any angle \theta, the sum of the square of the sine and the square of the cosine of that angle will always equal 1. This relationship is derived from the Pythagorean theorem, which relates the sides of a right-angled triangle. In a unit circle, where the radius is equal to 1, the sine of an angle represents the y-coordinate of a point on the circle's circumference, and the cosine represents the x-coordinate. The sum of their squares equals the square of the radius, hence proving the identity.
Understanding this identity can help greatly in verifying other more complex identities, such as:
\[\sin \theta \csc \theta-\sin^{2} \theta=\cos^{2} \theta.\]By recognizing the Pythagorean identity's part in this equation (\( 1 - \sin^2\theta \)), we quickly see the simplistic nature of verifying trigonometric identities. This foundational knowledge is essential in not only verifying identities but also in understanding the intrinsic properties of trigonometric functions.

Cosecant Function

The cosecant function, denoted as \(\csc\), is one of the six core trigonometric functions and is the reciprocal of the sine function. Essentially, for any angle \theta, the cosecant is defined as the ratio of the length of the hypotenuse to the length of the opposite side in a right-angled triangle, or:
\[\csc \theta = \frac{1}{\sin \theta}.\]Understanding the behavior of the \(\csc\) function is crucial when dealing with trigonometric identities. For example, in the identity:\(\sin \theta \csc \theta-\sin^{2} \theta=\cos^{2} \theta\), knowing that \(\csc \theta\) is \(\frac{1}{\sin \theta}\) allows you to simplify the left side of the equation, leading to an easier verification process.
Furthermore, complications with understanding the \(\csc\) function often arise because it's not as commonly used as sine, cosine, and tangent. Students sometimes forget that \(\csc\) can be graphed and analyzed like any other function, which brings us to how graphing utilities can help visualize and verify trigonometric identities involving the \(\csc\) function.

Graphing Utility for Trigonometry

Graphing utilities are incredibly helpful tools for visualizing functions and are particularly useful in trigonometry. They allow students to clearly see how trigonometric functions behave, and they can be instrumental in verifying trigonometric identities. For example, if you're trying to verify the identity \(\sin \theta \csc \theta-\sin^{2} \theta=\cos^{2} \theta\), a graphing utility can show the functions on the same set of axes.
By plotting \(y = \sin \theta \csc \theta-\sin^{2} \theta\) and \(y = \cos^2\theta\), if both equations are indeed equivalent, the graphing utility will display identical graphs. This provides a visual confirmation of the identity's correctness, supplementing the algebraic proof.

• It's essential to be familiar with graphing utilities, especially when learning trigonometry, as they offer an interactive way to understand complex concepts.
• They also aid in error checking, allowing you to catch mistakes that might have been made during algebraic manipulations.
• Moreover, they can be used to explore the effects of changing variables in real-time, making abstract concepts more tangible.

Graphing utilities facilitate a deeper understanding of trigonometry by enabling students to visualize the relationships and identities they are learning about.