Problem 21 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \cot 312^{\circ} $$

Step-by-Step Solution

Answer

The cofunction is \(\tan 138^{\circ} \) and the value is approximately -0.8098.

Step 1: Identify the Cofunction

For trigonometric functions, the cofunction of \( \cot \theta \) is \( \tan(90^{\circ} - \theta) \). Thus, the cofunction of \( \cot 312^{\circ} \) is \( \tan(90^{\circ} - 312^{\circ}) \).

Step 2: Simplify the Angle

Calculate \( 90^{\circ} - 312^{\circ} \) to find the equivalent angle for the cofunction. This is \( 90^{\circ} - 312^{\circ} = -222^{\circ} \).

Step 3: Standard Angle Conversion

Convert \( -222^{\circ} \) into a positive angle by adding \( 360^{\circ} \): \(-222^{\circ} + 360^{\circ} = 138^{\circ}\). Thus, \( \tan(-222^{\circ}) = \tan(138^{\circ}) \) because tangent function is periodic with a period of \( 180^{\circ} \).

Step 4: Calculate the Cofunction Value

Now compute \( \tan(138^{\circ}) \) using a calculator or trigonometric table. The value is approximately -0.8098 when rounded to four decimal places.

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In simple terms, for any angle \( \theta \), the cotangent is given by:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• Alternatively, it can also be written in terms of sine and cosine as: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Cotangent is particularly useful when dealing with angles where the tangent value is known or easy to calculate. In the context of the exercise, \( \cot 312^{\circ} \) involves finding an equivalent angle using its cofunction, the tangent.

Tangent

The tangent function, denoted as \( \tan \), is directly related to the cotangent and is amongst the primary trigonometric functions. Tangent of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the adjacent side. Mathematically, it is expressed as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

In the solution, we use the property that the cofunction of \( \cot \theta \) is \( \tan(90^{\circ} - \theta) \). This relationship is used to transform cotangent values into tangent values by adjusting the angle, which simplifies calculations under certain conditions.

Angle Simplification

Simplifying angles is a crucial step in solving trigonometric problems. It involves transforming a given angle into an equivalent one that either falls into the primary range \((0^{\circ} - 360^{\circ})\) or simplifies calculations.

• In our case, \( 90^{\circ} - 312^{\circ} = -222^{\circ} \), which is a negative angle.
• To convert it to a positive angle, we add \( 360^{\circ} \), resulting in \( 138^{\circ} \).

This conversion is essential because calculator functions generally take angles in this primary range, making it easier to evaluate the trigonometric function accurately.

Periodic Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. Understanding this property helps in simplifying calculations and converting angles.

• For the tangent function, the period is \( 180^{\circ} \). This means \( \tan(\theta) = \tan(\theta + 180^{\circ}n) \) for any integer \( n \).

In the exercise, we used this property to equate \( \tan(-222^{\circ}) \) with \( \tan(138^{\circ}) \). Despite being seemingly different angles, the periodic nature ensures they share the same tangent value, hence providing an equivalent solution.