Problem 64 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin \theta+\cos \theta}{\cos \theta}=1+\tan \theta$$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sin \theta+\cos \theta}{\cos \theta} = 1+\tan \theta \).

Step 1: Start with the Left-Hand Side

We have the left-hand side (LHS) of the identity: \( \frac{\sin \theta + \cos \theta}{\cos \theta} \). To verify the identity, we need to transform this expression into the right-hand side (RHS) that is \( 1 + \tan \theta \).

Step 2: Break Down the Fraction

Split the fraction \( \frac{\sin \theta + \cos \theta}{\cos \theta} \) into two separate fractions: \( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\cos \theta} \).

Step 3: Simplify Each Term

Simplify each term individually. \( \frac{\sin \theta}{\cos \theta} = \tan \theta \) and \( \frac{\cos \theta}{\cos \theta} = 1 \).

Step 4: Combine the Terms

Combine the simplified terms: \( 1 + \tan \theta \).

Step 5: Verify the Identity

The transformed left-hand side now matches the right-hand side: \( 1 + \tan \theta \). Thus, the identity is verified.

Key Concepts

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry, representing ratios of sides in a right triangle. The sine function, denoted as \( \sin \theta \), gives the ratio of the side opposite angle \( \theta \) to the hypotenuse. The cosine function, denoted as \( \cos \theta \), represents the ratio of the adjacent side to the hypotenuse. Together, these functions help us describe the shape, angle, and rotation in triangles.

• \( \sin \theta \) is calculated as the opposite side over the hypotenuse: \( \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta \) is found by dividing the adjacent side by the hypotenuse: \( \frac{\text{adjacent}}{\text{hypotenuse}} \)

These functions are also periodic, repeating values every \( 2\pi \) radians, which corresponds to a full circle rotation.

Tangent Function

The tangent function links closely with sine and cosine. It is expressed as the ratio of the sine of an angle to the cosine of the same angle. Mathematically, it is represented as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This relationship reveals that when \( \theta \) approaches \( 90^\circ \) or \( \frac{\pi}{2} \) radians, the cosine value approaches zero, making the tangent function approach infinity. It is this characteristic that explains the vertical asymptotes found in the graph of the tangent function, where it approaches infinitely positive or negative values.

• Useful for understanding slopes in trigonometry
• Aids in describing angles in terms of vertical and horizontal components

The tangent's periodic nature is every \( \pi \) radians due to its singularity at angles where cosine equals zero.

Fraction Simplification

Simplifying fractions in trigonometry involves expressing complex trigonometric ratios in simpler, equivalent forms. In our exercise, we simplified \( \frac{\sin \theta + \cos \theta}{\cos \theta} \) by breaking it down into individual terms:

• Split into \( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\cos \theta} \)
• Simplify each to obtain \( \tan \theta + 1 \)

This fraction simplification allows us to transition complex expressions into identifiers that are easier to compute and understand. Recognizing how to split terms and simplify them using known identities is key to verifying trigonometric identities. Such techniques are especially helpful when working through exams and in evaluating expressions rapidly.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are pivotal in exploring the properties of circles and oscillations. Each function has unique curves and properties that make it versatile in mathematical and real-world applications.
Trigonometric functions are often used in:

• Geometry: Describing angles and side lengths in polygons
• Physics: Modelling waves, oscillations, and circular motions
• Engineering: Calculating forces, rotations, and analyzing periodic motions

Understanding these functions enables us to analyze structures from bridge constructions to electronic circuits. The ability to transform and apply these functions to solve various trigonometric identities, like the one in our exercise, is invaluable in simplifying complex problems.