Problem 67 - Precalculus

Question

Evaluate the following expressions exactly: $$\cot \left(-315^{\circ}\right)$$

Step-by-Step Solution

Answer

\( \cot(-315^\circ) = 1 \).

Step 1: Understand the Cotangent Function

The cotangent of an angle \( \theta \) is the reciprocal of the tangent function. Hence, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Before evaluating \( \cot(-315^\circ) \), consider that the cotangent function is periodic and better understood in its positive equivalent angle.

Step 2: Convert the Angle to a Positive Equivalent

Angles can be expressed differently based on their position within a circle. A negative angle moves clockwise, whereas a positive angle moves counterclockwise. To convert \( -315^\circ \) into a positive angle, add \( 360^\circ \): \(-315^\circ + 360^\circ = 45^\circ.\)

Step 3: Evaluate the Cotangent of the Positive Angle

Now calculate \( \cot(45^\circ) \). Since \( \tan(45^\circ) = 1 \), its reciprocal \( \cot(45^\circ) = \frac{1}{1} = 1 \). Therefore, \( \cot(-315^\circ) = \cot(45^\circ) = 1.\)

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics used to relate the angles and sides of triangles, particularly right-angled triangles. The six primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each function corresponds to specific ratios - typically involving the sides of a right-angled triangle.

• Sine and cosine focus on opposite and adjacent sides relative to the angle.
• Tangent is the ratio of sine to cosine, expressed as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
• Cotangent is the reciprocal of the tangent.

Understanding these functions requires recognizing their ratios and how they behave over standard angles, such as 0, 30, 45, 60, and 90 degrees. Their periodic nature also means these functions repeat their values in regular intervals, making them very useful in various applications, from architecture to physics and even computer graphics.

Angle Conversion

Angles can be expressed in both degrees and radians, and sometimes they need converting for calculations or interpretations. Specifically, converting negative angles to positive can simplify problems. A negative angle is measured clockwise from the positive x-axis, while a positive angle is measured counterclockwise. For example, converting \(-315^{\circ}\) requires us to focus on the circular nature of angles:

• Complete revolution in a circle is \(360^{\circ}\), meaning anything exceeding this can be reduced.
• By adding \(360^{\circ}\) to \(-315^{\circ}\), we find its positive equivalent.
• \(-315^{\circ} + 360^{\circ} = 45^{\circ}\).

Another conversion often used is converting between radians and degrees, with\(\pi\) radians equating to \(180^{\circ}\). Conversions assist in interpreting angles across different mathematical tasks, ensuring compatibility with various mathematical principles.

Reciprocal Identities

Reciprocal identities are relationships between trigonometric functions that involve flipping the numerator and denominator of another function. These identities are useful as they often simplify expressions and equations in trigonometry.

• The cotangent function, for instance, is defined as the reciprocal of the tangent function.
• This means \(\cot(\theta) = \frac{1}{\tan(\theta)}\).
• Similarly, the cosecant and secant functions are reciprocals:\(\csc(\theta) = \frac{1}{\sin(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

Knowing these reciprocal identities helps in simplifying complex trigonometric expressions and solving equations efficiently. They highlight the interrelationships between different trigonometric functions, which is particularly helpful when tackling advanced math problems or verifying trigonometric expressions.