Problem 18 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(\cot \theta=\frac{16}{11}\)

Step-by-Step Solution

Answer

\(\theta = \arctan \left(\frac{11}{16}\right)\), then compute using a calculator.

Step 1: Find the Reciprocal

Given \(\cot \theta = \frac{16}{11}\), the reciprocal of this is \(\tan \theta = \frac{11}{16}\) as cotangent and tangent are reciprocal to each other.

Step 2: Find the Angle

To find the angle, use the tangent inverse function. \(\theta = \arctan \left(\frac{11}{16}\right)\). Use a calculator to compute this.

Key Concepts

Cotangent

Cotangent is one of the fundamental trigonometric functions. It can be thought of as the sister function to tangent, but instead of dealing directly with the opposite and adjacent sides of a right triangle, it takes their reciprocal.

• Definition: Cotangent of angle \(\theta\) is defined as \(\cot \theta = \frac{\text{adjacent}}{\text{opposite}}\)
• Relationship with Sine and Cosine: \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)

Being a reciprocal function, cotangent becomes undefined when sine is zero, as this would lead to division by zero.

Tangent

The tangent function is perhaps one of the most well-known trigonometric functions. It expresses a unique relationship within a right triangle involving its angle and the sides.

• Definition: Tangent of angle \(\theta\) is \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)
• Connection with angles: In the unit circle, tangent relates to the slope of the line connecting the origin to a point on the circle.
• Reciprocal Characteristics: It is reciprocal to the cotangent, allowing one to switch representations, such as \(\tan \theta = \frac{1}{\cot \theta}\).

Tangent emerges frequently in problems requiring the determination of an unknown angle or a side in right triangles.

Inverse trigonometric functions

Inverse trigonometric functions allow us to work backwards from functions like tangent, sine, and cosine to determine their angles.

• Functionality: Given \(\tan \theta = \frac{11}{16}\), the inverse function \(\arctan\) tells us what \(\theta\) actually is.
• Application: Inverse functions are commonly used in real-world applications such as engineering and physics, where precise angle measurements are needed.

To find the angle, using a calculator is often necessary for non-standard angles, providing us angles in radians or degrees.

Angles

Angles are a fundamental part of trigonometry and hold great significance in measuring rotations and different proportions of circles.

• Measurement: Commonly measured in degrees or radians, a full circle equates to 360 degrees or \(2\pi\) radians.
• Usage: Angles are utilized in trigonometry to find unknown dimensions in various geometric structures.

In the exercise, finding the angle \(\theta\) through the \(\arctan\) function illustrates how trigonometric functions directly connect to angle measurement.

Reciprocal identities

Reciprocal identities play a crucial role in simplifying and solving trigonometric expressions. For instance, if you know the value of one function, its reciprocal value can provide additional insight or avenues to solve a problem.

• Fundamental Reciprocals:
  • Tangent and Cotangent: \(\tan \theta = \frac{1}{\cot \theta}\)
  • Sine and Cosecant: \(\csc \theta = \frac{1}{\sin \theta}\)
  • Cosine and Secant: \(\sec \theta = \frac{1}{\cos \theta}\)
• Simplification: Reciprocal identities are pivotal in breaking down complex expressions to simpler forms.

These identities are vital in both theoretical trigonometry and practical problem-solving applications.