Problem 44 - Algebra 2

Question

Express the first trigonometric function in terms of the second. $$ \sec \theta, \tan \theta $$

Step-by-Step Solution

Answer

The secant function sec(θ) in terms of tangent function tan(θ) is \(sec(θ) = \sqrt{1 + tan^2(θ)}\)

Step 1: Recall the definition of sec(θ)

Secant of θ, or sec(θ), is defined as the reciprocal of cos(θ), i.e., sec(θ) = 1/cos(θ).

Step 2: Formulate the Pythagorean Identity

Pythagorean Identity in trigonometry states that \(1 + tan^2(x) = sec^2(x)\). This relationship holds true for any angle x.

Step 3: Isolate sec(θ)

Rearrange the Pythagorean identity to solve for sec(θ). This can be done by taking the square root of both sides, which gives \[sec(θ) = \sqrt{1 + tan^2(θ)}\] Note: Secant can be positive or negative. The sign is determined by the quadrant in which the angle θ lies.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec(\theta) \), plays a vital role in trigonometry by providing the reciprocal of cosine. Essentially, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This definition helps in exploring and solving various trigonometric equations. It's crucial to understand that just like other trigonometric functions, the secant function is periodic, with a period of \(2\pi\), meaning its values repeat every \(2\pi\) radians.

• Key Property: Reciprocal of the cosine function
• Periodicity: Its values repeat every \(2\pi\) radians
• Function Range: Result depends on whether \(\cos(\theta)\) is positive or negative, as \(\sec(\theta)\) will also take those signs accordingly

When determining the value of \(\sec(\theta)\), always pay attention to the quadrant in which \(\theta\) lies. Cosine is positive in the First and Fourth quadrants. Thus, \(\sec(\theta)\) would be positive in these quadrants as well.

Tangent Function

Unlike the secant function, the tangent function, \( \tan(\theta) \), is defined as the ratio of sine to cosine, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The tangent function is also periodic with a period of \(\pi\), meaning it repeats every \(\pi\) radians.

• Key Property: Ratio of sine to cosine
• Periodicity: Repeats every \(\pi\) radians
• Function Behavior: Can take all real values as \(\theta\) varies

It's helpful to remember that when \(\cos(\theta) = 0\), the tangent function becomes undefined, leading to vertical asymptotes in its graph. These points typically occur at angles like \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc., where the cosine function equals zero.

Pythagorean Identities

Pythagorean identities are foundational relationships between sine, cosine, tangent, and secant functions. The Pythagorean identity relevant to expressing \( \sec(\theta) \) in terms of \( \tan(\theta) \) is \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

• Basic Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Derived Identity: \( 1 + \tan^2(\theta) = \sec^2(\theta) \)
• Usage: Useful for converting between different trigonometric functions

To isolate and express \( \sec(\theta) \), you can rearrange this identity as \( \sec(\theta) = \sqrt{1 + \tan^2(\theta)} \). This formula indicates the relationship between these functions and enables solving problems where these functions interrelate. It's crucial to consider the sign of \(\sec(\theta)\) based on the quadrant in which \(\theta\) lies, as secant can be either positive or negative.