Question

Find the value of \(\sec \theta\) if \(\cot \theta=\frac{3}{4}\) \(180^{\circ} \leq \theta \leq 270^{\circ}\)

Short answer

\(\text{sec} \theta = - \frac{5}{3}\).

Step-by-step solution

Understand the given information

Given \(\theta\) is in the interval \(180^{\circ} \leq \theta \leq 270^{\circ}\). This places \(\theta\) in the third quadrant where tangent and cotangent are positive, and sine and cosine are negative. The given value is \(\text{cot} \theta = \frac{3}{4}\).

Use the identity to find \(\tan \theta\)

We know that \(\text{cot} \theta = \frac{1}{\tan \theta}\), so \(\tan \theta = \frac{1}{\text{cot} \theta} \). Substituting the given value, we get \(\tan \theta = \frac{1}{\frac{3}{4}} = \frac{4}{3}\).

Use the Pythagorean identity to find \(\text{sec} \theta\)

The Pythagorean identity is \( \tan^2 \theta + 1 = \text{sec}^2 \theta \). \ Plug in \(\tan \theta = \frac{4}{3} \) to find \( \text{sec}^2 \theta \): \[ \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9}\] and \[ \frac{16}{9} + 1 = \text{sec}^2 \theta \].

Solve for \(\text{sec} \theta\)

Combine the terms to find \(\text{sec}^2 \theta\): \[ \frac{16}{9} + 1 = \frac{16}{9} + \frac{9}{9} = \frac{25}{9}\]. Thus, \[ \text{sec}^2 \theta = \frac{25}{9}\], and taking the square root gives: \(\text{sec} \theta = \sqrt{ \frac{25}{9} } \). Since \(\theta\) is in the third quadrant, \(\text{sec} \theta\) is negative: \[ \text{sec} \theta = - \frac{5}{3} \].

Key concepts

cotangent

The cotangent function is one of the fundamental trigonometric ratios. Cotangent, abbreviated as \(\text{cot} \theta\) is the reciprocal of the tangent function. Mathematically, \(\text{cot} \theta = \frac{1}{\tan \theta}\). This means if you know the value of tangent, you can easily find cotangent, and vice-versa. For example, in the given problem where \(\text{cot} \theta = \frac{3}{4} \), we can quickly find that \(\tan \theta = \frac{4}{3} \). Understanding how cotangent relates to other trigonometric functions is key in solving complex trigonometric problems.

secant

Secant, abbreviated \(\text{sec} \theta\), is another important trigonometric function. It is the reciprocal of the cosine function: \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\). This concept is essential since it helps bridge understanding between different trigonometric identities and functions. In the problem, we needed to find \(\text{sec} \theta\), starting from the given \(\text{cot} \theta\). We utilized Pythagorean identities and found that \(\text{sec} \theta\) was negative in the third quadrant, leading to \(\text{sec} \theta = - \frac{5}{3}\). Remember, the sign and value of \(\text{sec} \theta\) heavily depend on the quadrant in which the angle lies.

Pythagorean identity

One crucial identity in trigonometry is the Pythagorean identity. The main form of this identity relates tangent and secant through the equation: \( \tan^2 \theta + 1 = \text{sec}^2 \theta \). This identity is derived from the Pythagorean theorem applied in the unit circle. In our problem, we used this identity to find \(\text{sec} \theta\) after knowing \(\tan \theta\). Plugging in \( \tan \theta = \frac{4}{3} \), we computed:

• \(\tan^2 \theta = \frac{16}{9} \)
• Adding 1: \(\frac{16}{9} + 1 = \frac{25}{9} \)
• Taking the square root to get \( \text{sec} \theta = \frac{\text{5}}{\text{3}} \)

Since the angle was in the third quadrant, \( \text{sec} \theta \) was negative, resulting in \( \text{sec} \theta = - \frac{5}{3} \).

trigonometric quadrants

The coordinate system in trigonometry is divided into four quadrants. Each quadrant affects the sign of trigonometric functions differently:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, while cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, while sine and tangent are negative.

Given \( \theta \) is between 180° and 270°, it places \( \theta \) in the third quadrant. Here, the sine and cosine are negative, affecting \( \text{sec} \theta \) (since \(\text{sec} \theta = \frac{1}{\text{cos} \theta} \)). This results in a negative value for \( \text{sec} \theta \), which was key to solving our problem.