Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{\sec \theta}{\csc \theta}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \tan \theta \).

Step 1: Express in terms of sine and cosine

First, recall the definitions of secant and cosecant in terms of sine and cosine. We have \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Substitute these into the original expression: \( \frac{\sec \theta}{\csc \theta} = \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} \).

Step 2: Simplify the Fraction

To simplify \( \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} \), use the property of dividing fractions, which is to multiply by the reciprocal. Thus, \[ \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} = \frac{1}{\cos \theta} \times \frac{\sin \theta}{1} = \frac{\sin \theta}{\cos \theta} \].

Step 3: Simplify using known identities

Now, recognize that \( \frac{\sin \theta}{\cos \theta} \) is equivalent to \( \tan \theta \). Therefore, the entire expression simplifies to \( \tan \theta \).

Key Concepts

Sine and Cosine

When beginning with trigonometric identities, sine and cosine are two of the most fundamental functions. In a right-angled triangle, the sine of an angle \( \theta \) is the ratio of the length of the side opposite to \( \theta \) to the hypotenuse. Cosine, on the other hand, is the ratio of the length of the adjacent side to the hypotenuse. These definitions can be remembered with the mnemonic "SOH CAH TOA."

Sine and cosine are periodic functions, each with a period of \( 2\pi \). This means the function values repeat every \( 2\pi \) radians. They are often represented as:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These identities lay the groundwork for understanding more complex functions and operations, such as the calculation of tangent, secant, and cosecant, which are expressed in terms of sine and cosine.

Secant and Cosecant

Secant and cosecant are reciprocal trigonometric identities. They are derived from the basic functions cosine and sine, respectively. To express these in terms of sine and cosine:

• Secant is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \)

These relationships can help in simplifying complex trigonometric expressions as in the given exercise, where secant and cosecant are expressed using the basic functions. This transformation is often necessary when simplifying or solving trigonometric equations because it might make the relationships more apparent or easier to manipulate.

Understanding these reciprocal identities enriches your comprehension of the trigonometric circle and how angles relate to each other, providing a deeper insight into their interactions in mathematical problems.

Tangent

Tangent is another primary trigonometric function that relates to both sine and cosine. In a right-angled triangle, tangent of an angle \( \theta \) is expressed as the ratio of the sine of the angle to the cosine of the angle. Mathematically, it is represented as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This expression is particularly useful, as seen in the original exercise solution. Simplifying the expression \( \frac{\sec \theta}{\csc \theta} \) led to the form \( \frac{\sin \theta}{\cos \theta} \). Recognizing this as \( \tan \theta \) simplifies the problem significantly.

Tangent can also be understood as measuring the "steepness" of the angle in the unit circle, which corresponds to the slope of the line. Its periodicity is \( \pi \), which is different from sine and cosine, and it provides unique insights in solving trigonometric equations, especially when dealing with non-right triangles where the angles might not be straightforward.