Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \(\sec \theta+\tan \theta\)

Step-by-Step Solution

Answer

\(\sec \theta + \tan \theta = \frac{1 + \sin \theta}{\cos \theta}\)

Step 1: Recall Definitions

Recall the definitions of secant and tangent in terms of sine and cosine. We have \[ \sec \theta = \frac{1}{\cos \theta} \] and \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \].

Step 2: Substitute Definitions

Substitute the definitions of secant and tangent into the expression \[ \sec \theta + \tan \theta = \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} \].

Step 3: Combine Fractions

Since both terms have the common denominator of \(\cos \theta\), we can combine them into a single fraction: \[ \frac{1 + \sin \theta}{\cos \theta} \].

Step 4: Simplify the Expression

The expression \(\frac{1 + \sin \theta}{\cos \theta}\) is already simplified as there are no common factors to cancel out.

Key Concepts

Sine and Cosine

Understanding sine and cosine is essential as they are the fundamental trigonometric functions. Sine, denoted as \( \sin \theta \), represents the ratio of the opposite side to the hypotenuse in a right triangle. In the unit circle, it corresponds to the y-coordinate of the point where the terminal side of an angle intersects the circle.
Cosine, denoted as \( \cos \theta \), represents the ratio of the adjacent side to the hypotenuse. On the unit circle, it is the x-coordinate of the intersecting point.
Both sine and cosine are periodic functions with a period of \(2\pi\), which means they repeat every \(360^\circ\). They are pivotal in defining other trigonometric functions and in simplifying trigonometric expressions. Sine and cosine also have useful identities such as the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( \cos^2 \theta = 1 - \sin^2 \theta \)
• \( \sin^2 \theta = 1 - \cos^2 \theta \)

These identities allow for substituting and simplifying complex trigonometric expressions.

Secant and Tangent

Secant and tangent are other essential trigonometric functions but offer different perspectives and uses than sine and cosine. Secant, denoted as \( \sec \theta \), is the reciprocal of cosine. So, it is defined as \( \sec \theta = \frac{1}{\cos \theta} \). This means it represents the reciprocal of the x-coordinate in the unit circle. In terms of right triangles, it's the ratio of the hypotenuse to the adjacent side.
Tangent, denoted as \( \tan \theta \), is the ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Within a right triangle, it signifies the ratio of the opposite side to the adjacent side. On the unit circle, it reflects the slope of the line formed from the origin to the point on the circle.
These functions are vital:

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

Remembering these definitions helps in transforming and simplifying trigonometric expressions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill in trigonometry, helping to convert complex expressions into simpler, more manageable forms. Such expressions often involve a combination of various trigonometric functions.
To simplify them, one common technique is to convert all terms to sine and cosine since these are the fundamental trigonometric functions. Using their definitions and identities can make combining terms easier.
In our original problem, to simplify \( \sec \theta + \tan \theta \), we first express secant and tangent in terms of sine and cosine:

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

Substituting yields \( \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} \).
Since both terms have a common denominator, \( \cos \theta \), they can be combined into \( \frac{1 + \sin \theta}{\cos \theta} \). This is the expression in its simplest form because no further factors can be canceled.
Always seek common denominators and simplify wherever possible.