Problem 17 - 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. $$ \tan 237^{\circ} $$

Step-by-Step Solution

Answer

\( \tan 237^{\circ} = \cot(-147^{\circ}) \approx 2.1445 \).

Step 1: Identifying the Cofunction

The problem requires identifying the cofunction of \( \tan 237^{\circ} \). The cofunction identity for tangent is given by \( \tan(90^{\circ} - \theta) = \cot \theta \). Hence, the cofunction of tangent is cotangent.

Step 2: Rewriting Tangent in Terms of its Cofunction

For \( \tan 237^{\circ} \), we use the identity for cofunctions: \( \tan(\theta) = \cot(90^{\circ} - \theta) \). We determine that \( \theta = 237^{\circ} \) and therefore, the expression in cofunction terms is \( \cot(90^{\circ} - 237^{\circ}) \).

Step 3: Calculating the Angle for the Cofunction

We calculate the angle for the cofunction by subtracting 237 from 90: \( 90^{\circ} - 237^{\circ} = -147^{\circ} \). Thus, \( \tan 237^{\circ} = \cot(-147^{\circ}) \).

Step 4: Finding the Value of the Cofunction

Now, compute \( \cot(-147^{\circ}) \). Use the identity \( \cot(x) = \frac{1}{\tan(x)} \). First, find \( \tan(-147^{\circ}) \) and then take its reciprocal to find \( \cot(-147^{\circ}) \).

Step 5: Using Calculators for Decimals

Use a calculator to find \( \tan(-147^{\circ}) \). \( \tan(-147^{\circ}) \approx 0.4663 \), hence, \( \cot(-147^{\circ}) = \frac{1}{0.4663} \approx 2.1445 \). Round this to four decimal places.

Key Concepts

tangent function

The tangent function, often abbreviated as "tan," is a fundamental trigonometric function which is usually associated with right triangles. It is particularly useful in describing the ratio of two sides of a triangle - the opposite side to the adjacent side. In a formula, it can be expressed as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

This simple relation helps solve many problems involving angles and lengths in geometry and physics as well.
In addition to its basic definition, the tangent function has an important angle property: it is periodic with a period of \(180^{\circ}\) or \(\pi\) radians. This means that the function repeats its values every \(180^{\circ}\).
Additionally, the tangent of an angle is undefined at odd multiples of \(90^{\circ}\) because at these angles, the adjacent side of the triangle becomes zero, which makes the denominator in the fraction \( \frac{\text{opposite}}{\text{adjacent}} \) zero; thus, division by zero is not possible.

cotangent function

The cotangent function, abbreviated as "cot," is the reciprocal of the tangent function. In simpler terms, it is defined based on the ratio of the adjacent side to the opposite side of a right triangle, doing essentially the opposite of the tangent. It can be mathematically expressed by:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \)

Understanding the relationship between tangent and cotangent is important as they are cofunctions of each other.
The cotangent function is also periodic but, like tangent, its period is \(180^{\circ}\) or \(\pi\) radians. This means that the graph of cotangent repeats every \(180^{\circ}\), aligning with tangent's behavior.
Since cotangent is a reciprocal of tangent, it is undefined where tangent equals zero. This happens at integer multiples of \(180^{\circ}\), resulting in points where the function cannot be evaluated.

angle subtraction

Angle subtraction in trigonometry often involves determining the result of subtracting one angle from another, especially in relation to cofunctions. The cofunction identities help establish the relationships between angles. Given two angles, the difference \(A - B\) can often be used in the form of trigonometric identities to simplify expressions.
When solving trigonometric problems, especially those utilizing cofunction identities, angle subtraction is pivotal. For example, to convert an angle into a cofunction using the identity \( \tan(90^{\circ} - \theta) = \cot(\theta) \), one needs to calculate \(90^{\circ} - \theta\). This simple subtraction gives the angle needed to express tangent in terms of cotangent.

• Example: Subtract the angle \(237^{\circ}\) from \(90^{\circ}\): \[ 90^{\circ} - 237^{\circ} = -147^{\circ} \]

This resulting negative angle can then be evaluated using standard trigonometric operations or with the help of a calculator for more accuracy.