Problem 42 - Algebra 2

Question

Evaluate each expression. Write your answer in exact form. If appropriate, also state it as a decimal rounded to the nearest hundredth. If the expression is undefined, write undefined. $$ \cot \left(-45^{\circ}\right) $$

Step-by-Step Solution

Answer

\(\cot(-45^{\circ}) = -1\)

Step 1 - Understand the Cotangent Function

Cotangent of an angle in a right-angled triangle is the ratio of the adjacent side to the opposite side. It is also defined as the reciprocal of the tangent function, i.e. \(\cot x = \frac{1}{\tan x}\).

Step 2 - Evaluate the Tangent

First, the tangent of -45 degrees needs to be found to compute the cotangent. It is known that \(\tan(-\theta) = -\tan(\theta)\). As the \(\tan(45^{\circ})\) is 1, the \(\tan(-45^{\circ})\) would be -1.

Step 3 - Calculate the Cotangent

Finally, as the cotangent is the reciprocal of tangent, it can be evaluated as \(\cot(-45^{\circ}) = -\cot(45^{\circ}) = -1\). Therefore, the exact value of \(\cot(-45^{\circ})\) is -1.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle. They relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions include sine, cosine, and tangent. In addition, we have the reciprocal functions which are cosecant, secant, and cotangent. These functions are vital for analyzing right-angled triangles and for solving problems involving periodic phenomena such as sound and light waves.

• Sine (\(\sin\theta\) relates the opposite side to the hypotenuse.
• Cosine (\(\cos\theta\) relates the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\) is the ratio of the opposite side to the adjacent side.
• Cotangent (\(\cot\theta\)) is the reciprocal of the tangent function, defined as the adjacent side over the opposite side.

Understanding these functions helps in solving trigonometric problems efficiently. The cotangent function, specifically, is useful when the relationship between the adjacent and opposite sides of a triangle is of interest.

Angle Measurements

In trigonometry, angles are typically measured in two ways: degrees and radians. In the context of solving trigonometric expressions like \(\cot (-45^{\circ})\), understanding angle measurement in degrees is necessary.
- Degrees: This is the most common measure of angles in everyday contexts. A full circle has 360 degrees. - Radians: Radians are another unit of angle measurement commonly used in calculus. One complete circle is \(2\pi\) radians.
To convert between radians and degrees, use the relations:

• \(1\,\text{degree} = \frac{\pi}{180}\,\text{radians}\)
• \(1\,\text{radian} = \frac{180}{\pi}\,\text{degrees}\)

For the specific case of \(-45^{\circ}\), understanding its place on the unit circle is crucial for correctly utilizing reciprocal identities and evaluating trigonometric functions.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that relate trigonometric functions to their reciprocals. For example, cotangent is the reciprocal of tangent, written as \(\cot x = \frac{1}{\tan x}\) . Let's look at how these identities work:

• Cosecant (\(\csc\theta\)): Reciprocal of sine, \(\csc\theta = \frac{1}{\sin\theta}\).
• Secant (\(\sec\theta\)): Reciprocal of cosine, \(\sec\theta = \frac{1}{\cos\theta}\).
• Cotangent (\(\cot\theta\)): Reciprocal of tangent, \(\cot\theta = \frac{1}{\tan\theta}\).

Understanding these identities is crucial, especially when solving equations involving angle transformations or dealing with negative angles like \(-45^{\circ}\). By computing \(\tan(-45^{\circ}) = -1\), and applying the reciprocal identity, the cotangent becomes \(-1\). Thus reverting to the reciprocal identity simplifies solving.

Right-angled Triangles

Right-angled triangles are a foundation of trigonometry. They consist of one angle that is 90 degrees, making calculations straightforward by allowing specific trigonometric functions to be directly applied. In every right-angled triangle:

• Hypotenuse: The longest side of the triangle opposite the right angle.
• Opposite side: The side opposite the angle of interest.
• Adjacent side: The side next to the angle of interest.

When dealing with functions like the cotangent, in a right-angled triangle context, \(\cot(\theta)\) describes the ratio of the length of the adjacent side to the opposite side.
Utilizing these concepts helps in understanding the tangent and cotangent functions and how they can be evaluated, such as finding the value of \(\tan(-45^{\circ}) = -1\), and subsequently, \(\cot(-45^{\circ})= -1\). In practice, understanding these sides and functions simplifies complex problem solving in trigonometry.