Problem 108 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. $$\sin \theta \csc \theta=1$$

Step-by-Step Solution

Answer

Yes, the statement \( \sin \theta \csc \theta = 1 \) is true for all values of \( \theta \) not equal to zero.

Step 1: Understanding the given equation

The given equation states that the product of \( \sin \theta \) and \( \csc \theta \) equals 1.

Step 2: Express in terms of sine

Recall that \( \csc \theta \) is the reciprocal of \( \sin \theta \), it can be written as \( \csc \theta = \frac{1}{\sin \theta} \). Then substitute this into the given equation to obtain \( \sin \theta \times \frac{1}{\sin \theta} \).

Step 3: Simplify the equation

After multiplying \( \sin \theta \) with \( \frac{1}{\sin \theta} \), we get the result as 1.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that analyzes the relationship between the angle in a right triangle and the ratio of its opposite side to the hypotenuse. - The sine of an angle, represented as \( \sin \theta \), indicates how much the angle tilts or inclines. The value of \( \sin \theta \) ranges from -1 to 1.- In the unit circle, where the radius is one, \( \sin \theta \) equals the y-coordinate of the point where the terminal side of the angle intersects the circle. For certain key angles, you might memorize specific sine values:

• \( \sin 0 = 0 \)
• \( \sin \left(\frac{\pi}{2}\right) = 1 \)
• \( \sin \pi = 0 \)
• \( \sin \left(\frac{3\pi}{2}\right) = -1 \)
• \( \sin 2\pi = 0 \)

Understanding the sine function enables you to solve a variety of problems in trigonometry and aids in comprehending reciprocal relationships like those with the cosecant.

Cosecant Function

The cosecant function is the reciprocal of the sine function. It helps describe aspects of a right triangle that are complementary to the sine function. - Mathematically, the cosecant of an angle \( \theta \) is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).- This function becomes undefined when \( \sin \theta = 0 \), as division by zero is not possible. Thus, \( \csc \theta \) is defined only when \( \theta \) is not an integral multiple of \( \pi \) (where sine equals zero).In real-world contexts, the cosecant function might not be as frequently used as sine, but it provides a valuable tool for solving advanced trigonometric equations. Using a reciprocal identity such as this shifts emphasis from understanding the slope or y-value (as sine does) to looking at how that value reciprocates on the unit circle.

Reciprocal Identity

Reciprocal identities are essential in trigonometry because they connect various trigonometric functions through their reciprocals. - A reciprocal identity involves expressing one trigonometric function as the reciprocal of another, as seen with \( \csc \theta = \frac{1}{\sin \theta} \).- These identities are helpful in simplifying complex trigonometric expressions and solving equations. In any trigonometric identity, knowing one aspect means you can deduce others.For example:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

When using the reciprocal identity \( \csc \theta = \frac{1}{\sin \theta} \), multiplying \( \sin \theta \cdot \csc \theta \) results in 1, thus verifying identities like those in the example equation \( \sin \theta \cdot \csc \theta = 1 \). This concept explains how reciprocal functions interact, proving them mathematically correct in given contexts.