Problem 29 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sin \theta}{\tan \theta}=\cos \theta $$

Step-by-Step Solution

Answer

The identity is verified; both sides equal \( \cos \theta \).

Step 1: Understanding the Identity

We need to verify the trigonometric identity: \( \frac{\sin \theta}{\tan \theta} = \cos \theta \). This means we must show that the left-hand side (LHS) is mathematically equal to the right-hand side (RHS).

Step 2: Rewriting Tangent in Terms of Sine and Cosine

Recall the trigonometric identity for tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). By using this identity, we can rewrite \( \tan \theta \) in the expression.

Step 3: Substitute into the Expression

Replace \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) in the LHS: \( \frac{\sin \theta}{\tan \theta} = \frac{\sin \theta}{\frac{\sin \theta}{\cos \theta}} \).

Step 4: Simplify the Fraction

Simplify the complex fraction by multiplying by the reciprocal: \( \frac{\sin \theta}{\frac{\sin \theta}{\cos \theta}} = \sin \theta \cdot \frac{\cos \theta}{\sin \theta} \).

Step 5: Cancel Out Terms

Cancel the \( \sin \theta \) terms: \( \frac{\sin \theta}{\sin \theta} = 1 \), thus we are left with \( \cos \theta \).

Step 6: Verification

The LHS simplifies to \( \cos \theta \), which is the same as the RHS. Therefore, the identity \( \frac{\sin \theta}{\tan \theta} = \cos \theta \) is verified.

Key Concepts

Sine Function

The sine function, often abbreviated as "sin," is a fundamental concept in trigonometry. It is one of the primary trigonometric functions that relates to the ratios of the sides of a right triangle. Specifically, the sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. This can be expressed as:
\[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]
In the unit circle, where the hypotenuse is always 1, the sine of an angle θ represents the y-coordinate of a point on the circle's circumference. This geometrical interpretation helps in understanding trigonometric functions beyond just right-angle scenarios. Here are some key points to remember:

• The sine function is periodic with a period of \(2\pi\).
• Sine values range between -1 and 1.
• The sine function is positive in both the first and second quadrants of the unit circle.

Cosine Function

The cosine function, abbreviated as "cos," is another essential trigonometric function that complements the sine function. It describes the relationship between the adjacent side of a right triangle to its hypotenuse:
\[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
On the unit circle, the cosine of an angle θ is represented by the x-coordinate of a point on the circle. This makes it intuitive to visualize on the unit circle as it varies with θ. Important features of the cosine function include:

• It is periodic with a period of \(2\pi\), just like the sine function.
• Cos values also oscillate between -1 and 1.
• The cosine function is positive in the first and fourth quadrants.

Understanding the cosine function is crucial when verifying trigonometric identities, like in the given exercise where we verified \( \frac{\sin \theta}{\tan \theta} = \cos \theta \). This emphasizes the interplay between sine and cosine through the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Tangent Function

The tangent function, symbolized as "tan," provides a comprehensive view of trigonometric relationships when combined with sine and cosine. It is defined as the ratio of the sine function to the cosine function for a given angle:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This identity is particularly useful for converting expressions in trigonometry. For example, in our original exercise, we used this to simplify and eventually verify the identity \( \frac{\sin \theta}{\tan \theta} = \cos \theta \). It's beneficial to note the following about the tangent function:

• Tangent is periodic with a period of \(\pi\), unlike sine and cosine.
• The function has no range limits, meaning its values can extend to both positive and negative infinity as it approaches its vertical asymptotes.
• Tangent is positive in the first and third quadrants, reflecting the sign changes of sine and cosine.

In terms of practical use, understanding tangent helps in solving triangles and verifying identities, as demonstrated in the step-by-step solution.