Problem 52 - 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.) $$\frac{\cos \theta+\sin \theta}{\sin \theta}$$

Step-by-Step Solution

Answer

The simplified expression is \( \cot \theta + 1 \).

Step 1: Express cosine over sine

Write the expression \( \frac{\cos \theta + \sin \theta}{\sin \theta} \) as sum of two fractions by separating the terms in the numerator: \[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\sin \theta} \].

Step 2: Simplify Each Fraction

For \( \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\sin \theta} \), simplify each fraction individually. You have: \[ \frac{\cos \theta}{\sin \theta} = \cot \theta \] and \[ \frac{\sin \theta}{\sin \theta} = 1 \].

Step 3: Combine Simplified Terms

Add the simplified terms from Step 2: \( \cot \theta + 1 \). This is your final expression.

Key Concepts

Understanding Sine

Sine is an essential trigonometric function that plays a crucial role in understanding and solving many mathematical problems, particularly those involving triangles and wave forms. It is defined based on a right-angled triangle or the unit circle.

• In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Using the unit circle, for an angle \( \theta \), the sine value is the y-coordinate of the point where the terminal side of the angle intersects the circle.

Sine is periodic and oscillates between -1 and 1 as the angle increases, creating a wave-like pattern. This function is pivotal when breaking down other trigonometric expressions, as seen in our original exercise. By translating complex expressions into sine terms, one can often simplify and solve them more easily, paving the way for deeper understanding.

Decoding Cosine

Cosine, another fundamental trigonometric function, complements sine in describing the properties of angles and triangles. Like sine, cosine is easy to define mathematically.

• In a right-angled triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• On the unit circle, cosine corresponds to the x-coordinate of the point on the circle at a given angle \( \theta \).

Cosine shares properties with sine, such as periodicity and bounded values, ranging from -1 to 1. Understanding cosine is critical because it often pairs with sine for various mathematical transformations and derivations. In the context of the problem, cosine must be articulated clearly to allow the expression to be rewritten for simplification.

Exploring Cotangent

Cotangent might not be as widely familiar as sine or cosine, but it is equally important in trigonometry. It is defined as the reciprocal of the tangent function and further helps in simplifying complex trigonometric expressions.

• For a given angle \( \theta \), cotangent is expressed as the ratio of the cosine to the sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• It is primarily used in substituting tangent for certain types of trigonometric simplifications and calculations.

In the exercise discussed, recognizing the cotangent function proves vital. By doing so, the initial problem reduces to a simpler expression in the form of \( \cot \theta + 1 \). This demonstrates how understanding trigonometric identities, like cotangent, helps transform and resolve complex equations efficiently.