Problem 80 - Algebra 2

Question

For Exercises \(79-82,\) suppose tan \(\theta=\frac{4}{3}, \sin \theta>0,\) and \(-\frac{\pi}{2} \leq \theta<\frac{\pi}{2}\) . Enter each answer as a decimal. What is \((\sin \theta)(\cot \theta) ?\)

Step-by-Step Solution

Answer

\((\sin \theta)(\cot \theta) = \frac{3}{5}\)

Step 1: Find \(sin \theta\)

Use the identity \(\tan \theta = \frac{sin \theta}{cos \theta}\) to solve for \(sin \theta\). Here, \(tan \theta = \frac{4}{3}\), so we have \(\sin \theta= \frac{4}{\sqrt{4^2 + 3^2}} = \frac{4}{5}\).

Step 2: Find \(cot \theta\)

Since cotangent is the reciprocal of tangent, we have \(cot \theta = \frac{1}{tan \theta} = \frac{3}{4}\).

Step 3: Calculate \((\sin \theta)(\cot \theta)\)

Use the values from Step 1 and Step 2 to find \((\sin \theta)(\cot \theta) = (\frac{4}{5})(\frac{3}{4}) = \frac{3}{5}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships in right triangles. These functions include sine \( \sin \theta \), cosine \( \cos \theta \), and tangent \( \tan \theta \), amongst others. They are used extensively in various fields like physics, engineering, and even in technology.

• The functions are based on ratios of sides in a right triangle.
• Each function deals with different side combinations related to the angle.

Understanding these ratios helps to solve many types of mathematical problems involving angles.

Tangent and Cotangent

Tangent \( \tan \theta \) and cotangent \( \cot \theta \) are related trigonometric functions. Tangent is defined as the ratio of the opposite side to the adjacent side for a given angle in a right triangle.

• In mathematical terms: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Cotangent is the reciprocal of tangent.

• This means \( \cot \theta = \frac{1}{\tan \theta} \). In terms of the triangle, it's the ratio of the adjacent side to the opposite side.

These functions are helpful for solving problems where you need to understand the relationships between angles and sides.

Sine and Cosine

Sine \( \sin \theta \) and cosine \( \cos \theta \) are the primary trigonometric functions. They describe the relationship between an angle and the lengths of the sides of a right triangle.

• Sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse: \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
• Cosine is the ratio of the length of the adjacent side to the hypotenuse: \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

These functions allow us to calculate unknown values in triangles and are often used in waveform analysis and oscillating movements.

Angle Measures

Angles can be measured in degrees or radians, and being comfortable with these measures is essential in trigonometry. Radians are a more natural way to express angles in terms of \pi \, which helps in mathematical calculations.

• One complete revolution around a circle is \( 2\pi \) radians (equal to 360 degrees).
• The interval given \( -\frac{\pi}{2} \leq \theta < \frac{\pi}{2} \) indicates that \( \theta \) is within the first and fourth quadrants, where sine is positive.

Understanding these concepts helps you interpret and solve trigonometric problems more easily.