Problem 41 - Algebra 2

Question

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cot \frac{\theta}{2} $$

Step-by-Step Solution

Answer

\(\cot{\frac{\theta}{2}} = -2\)

Step 1: Determine the sine of the given angle

Since the given angle \(\theta\) lies in the fourth quadrant where cos is positive and sin is negative, we can use the Pythagorean identity \(sin^2{\theta} + cos^2{\theta} = 1\). Solving for \(\sin{\theta}\) we find: \(\sin{\theta} = -\sqrt{1 - cos^2{\theta}} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\frac{4}{5}\).

Step 2: Calculate the cotangent of half of given angle

Utilizing the half-angle formula for cot: \(\cot{\frac{\theta}{2}} = \frac{1+\cos{\theta}}{\sin{\theta}}\), calculation gives: \(\cot{\frac{\theta}{2}} = \frac{1+ \frac{3}{5}}{-\frac{4}{5}} = -\frac{8}{4} = -2\) which is the exact value of the cotangent of half the given angle.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding the relationships between angles and sides in right triangles. These functions include sine, cosine, tangent, and others such as cotangent, secant, and cosecant.

Each function has a specific meaning:

• Sine (\( ext{sin}\)) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( ext{cos}\)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)) is the ratio of the opposite side to the adjacent side.
• Cotangent (\( ext{cot}\)) is the reciprocal of tangent, or adjacent over opposite.

By understanding these basic relationships, you can solve various problems about angles and their measures, particularly in trigonometry and calculus. The functions extend beyond simple right triangles and provide insight into periodic phenomena, such as waves and oscillations.

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry. It ties together the squares of sine and cosine, forming the equation:\[sin^2 \theta + cos^2 \theta = 1\]This identity is rooted in the Pythagorean theorem, which explains why it's so widely applicable. In any given situation where you have one of these values (like \( \cos \theta\)), you can find the other (like \( \sin \theta\)) using this identity.

For instance, in our problem, we know \( \cos \theta = \frac{3}{5} \). By applying the identity:

• Calculate \( sin^2 \theta = 1 - (\frac{3}{5})^2 = \frac{16}{25} \).
• Hence, \( \sin \theta = -\frac{4}{5} \) because sine is negative in the fourth quadrant.

This relationship simplifies calculations and is indispensable across different trigonometric problems.

Half-Angle Formulas

Half-angle formulas are tools in trigonometry that allow us to find the value of trigonometric functions at half an angle, given the original angle. These formulas can be especially useful when dealing with complex angles.

The half-angle formula for finding the cotangent of half an angle \( \theta \) is:\[ \cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta}\]Let's use this formula in our problem:

• Given \( \cos \theta = \frac{3}{5} \) and \( \sin \theta = -\frac{4}{5} \).
• Plug into the formula: \( \frac{1 + \frac{3}{5}}{-\frac{4}{5}} \).
• Simplify: \( \cot \frac{\theta}{2} = -2 \).

Half-angle formulas help bridge the gap between angles and their halves, making computations easier in various mathematical contexts.

Quadrant Analysis

Understanding which quadrant an angle falls into is vital for correctly determining the signs of the trigonometric functions. The quadrants in the coordinate plane are numbered counterclockwise from the positive x-axis.

Each quadrant affects the sign of the trigonometric functions:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

In our exercise, with \( 270^\circ < \theta < 360^\circ \), the angle is in the fourth quadrant. This means:

• Cosine is positive.
• Sine is negative.
• Tangent and cotangent are negative.

By performing a quadrant analysis, we appropriately determine the signs and values of trigonometric functions, ensuring the accuracy of our calculations.