Problem 22 - 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. \(\sec \theta(1+\cot \theta)-\csc \theta(1+\tan \theta)\)

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Rewrite Trigonometric Identities

Rewrite the given expression using basic trigonometric identities: \( \sec \theta = \frac{1}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 2: Substitute Identities into Expression

Substitute the identities from Step 1 into the original expression: \( \frac{1}{\cos \theta} (1 + \frac{\cos \theta}{\sin \theta}) - \frac{1}{\sin \theta} (1 + \frac{\sin \theta}{\cos \theta}) \).

Step 3: Simplify Each Term

Simplify each term separately:1. \( \frac{1}{\cos \theta} (1 + \frac{\cos \theta}{\sin \theta}) = \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \).2. \( \frac{1}{\sin \theta} (1 + \frac{\sin \theta}{\cos \theta}) = \frac{1}{\sin \theta} + \frac{1}{\cos \theta} \).

Step 4: Combine and Simplify the Expression

Combine the simplified terms from Step 3:\( \left( \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \right) - \left( \frac{1}{\sin \theta} + \frac{1}{\cos \theta} \right) \).

Step 5: Final Simplification

Notice that \( \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \) cancels out with \( \frac{1}{\sin \theta} + \frac{1}{\cos \theta} \). Therefore, the entire expression simplifies to \( 0 \).

Key Concepts

Sine and Cosine

When discussing trigonometric functions, sine and cosine are two of the most fundamental and widely used. These functions relate the angles of a right triangle to the lengths of its sides.
Sine, abbreviated as \( \sin \theta \), is the ratio of the length of the side opposite to the angle \( \theta \) to the hypotenuse of the triangle. In mathematical terms, it is defined as:

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

Cosine, or \( \cos \theta \), is a similar function, but it measures the ratio of the adjacent side of the angle \( \theta \) to the hypotenuse:

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

Both sine and cosine functions are essential in defining other trigonometric identities and relations. For example, any angle \( \theta \) can be converted into sine and cosine expressions, which helps to simplify more complex trigonometric expressions, just as in the given example exercise.
Sine and cosine values repeat in a cyclic pattern, with each cycle being 360 degrees or \( 2\pi \) radians. Understanding these functions is crucial for grasping more advanced trigonometry and calculus concepts.

Secant and Cosecant

Secant and cosecant are less common trigonometric functions compared to sine and cosine, but they are equally vital in understanding trigonometric identities.
The secant function is the reciprocal of the cosine function. It relates to horizontal distances in a unit circle. Mathematically expressed, secant is:

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

Cosecant is similarly the reciprocal of sine, relating to vertical distances in a unit circle:

• \( \csc \theta = \frac{1}{\sin \theta} \)

These functions are particularly useful in transformations and simplification of trigonometric equations. In our provided exercise, converting secant and cosecant into expressions with sine and cosine allowed for efficient simplification.
These reciprocal identities are often employed to handle computations involving angles greater than 90 degrees, or to simplify complex trigonometric expressions just like the one we evaluated. Recognizing and transforming these functions into sine and cosine can often reveal more straightforward calculations and insights.

Cotangent and Tangent

Cotangent and tangent are another pair of trigonometric functions, which differ in how they relate to a right triangle's sides compared to sine and cosine.
The tangent of an angle \( \theta \) represents the ratio of the sine of the angle to the cosine of the angle:

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

This function connects the opposite and adjacent sides of a right triangle without needing the hypotenuse. In contrast, cotangent is the reciprocal of tangent, relating adjacent side to opposite side:

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

In the context of simplifying trigonometric expressions, these identities can be articulated in terms of sine and cosine, as seen in the given solution where \( \cot \theta \) and \( \tan \theta \) were rewritten using their fundamental ratios.
The tangent and cotangent cycles are important for solving many geometric and trigonometric problems, as their periodic nature every \( \pi \) radians, or 180 degrees, often leads to solutions and simplifications similar to our exercise.