Problem 39 - Algebra and Trigonometry

Question

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \tan \theta, \quad \cos \theta ; \quad \theta \text { in Quadrant III } $$

Step-by-Step Solution

Answer

\( \tan \theta = -\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \) in Quadrant III.

Step 1: Understand the Relationship

Realize that in trigonometry, the tangent function can be expressed in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Hence, we need to express both sine and tangent in terms of cosine.

Step 2: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Since we need \( \sin \theta \), rearrange to get \( \sin^2 \theta = 1 - \cos^2 \theta \).

Step 3: Determine the Sign in Quadrant III

In Quadrant III, both sine and cosine are negative. Therefore, \( \sin \theta = -\sqrt{1 - \cos^2 \theta} \).

Step 4: Substitute into Tangent Expression

Substitute \( \sin \theta = -\sqrt{1 - \cos^2 \theta} \) into the tangent expression: \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Step 5: Simplify the Expression

The expression for \( \tan \theta \) in terms of \( \cos \theta \) in Quadrant III simplifies to \( \tan \theta = -\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Key Concepts

Tangent Identity

In trigonometry, one of the fundamental identities involves the tangent function. The tangent identity is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means that tangent is the ratio of the sine function to the cosine function.
When dealing with exercises that ask to express tangent in terms of cosine, we must consider how sine relates to cosine. This identity is crucial because it allows us to express one trigonometric function through others, facilitating various simplifications in solving trigonometric equations.

• Helps in connecting different trigonometric functions
• Useful in simplifying complex expressions
• Enables solution of equations by expressing unknowns in terms of known functions

Cosine Function

The cosine function, written as \( \cos \theta \), is one of the three primary trigonometric functions. It represents the x-coordinate of a point on the unit circle that is formed by rotating a ray by an angle \( \theta \).
In any application, knowing the value of \( \cos \theta \) is essential since it serves as a building block for defining other trigonometric functions like tangent and sine in multiple ways. In this exercise, cosine simplifies the expression of tangent by being the base for writing it in a more workable form.

• Fundamental function alongside sine and tangent
• Expresses relationships between angles and sides in right triangles
• Helps in transforming and solving trigonometric identities

Quadrant III

Angles in trigonometry are organized into four quadrants, and Quadrant III is where both sine and cosine are negative. This has implications for all associated trigonometric functions.
In Quadrant III, because both sine and cosine are negative,

• \( \tan \theta \) becomes positive since it's the ratio of two negative numbers.
• This knowledge is key when solving problems that ask for the sign of a trigonometric function, impacting the outcome of trigonometric identities.

When determining values in Quadrant III, it is critical to remember that the cosine function \( \cos \theta \) and the sine function \( \sin \theta \) need to maintain their respective negative signs.

Pythagorean Identity

The Pythagorean Identity is a fundamental principle in trigonometry. It states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This relationship is derived from the Pythagorean theorem and serves as a backbone for many trigonometric identities.
In our problem, using the Pythagorean identity allows the expression of \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
This equation enables the isolation and determination of \( \sin \theta \) from \( \cos \theta \), further enabling us to substitute into the tangent expression to rewrite \( \tan \theta \) in terms of \( \cos \theta \).

• Helps transform trigonometric expressions
• Facilitates solving for one function in terms of another
• Used universally in calculus and geometric applications