Problem 48 - Precalculus : Building Concepts and Connections

Question

Find a cofunction that has the same value as the given quantity. $$\tan 20^{\circ}$$

Step-by-Step Solution

Answer

The cofunction that has the same value as \( \tan 20^{\circ} \) is \( \cot 70^{\circ} \).

Step 1: Find the Complementary Angle

Firstly, we need to find the complement of \( 20^{\circ} \). The complement of an angle is simply \( 90^{\circ} \) minus that angle. So, the complement of \( 20^{\circ} \) is \( 90^{\circ}-20^{\circ}=70^{\circ} \).

Step 2: Determine the Cofunction

The cofunction of the tangent function is the cotangent function (cot). Thus, to find the cofunction that has the same value as \( \tan 20^{\circ} \), we find \( \cot \) of the complement of \( 20^{\circ} \). Hence, the cofunction is \( \cot 70^{\circ} \).

Key Concepts

Cofunction Identities

Cofunction identities are foundational concepts in trigonometry. They reveal a special relationship between certain trigonometric functions and their complements. Here is how they work: if you have an angle \( \theta \), its complementary angle is \( 90^{\circ} - \theta \). Two main cofunction pairings to remember are:

• \( \sin \theta = \cos (90^{\circ} - \theta) \)
• \( \cos \theta = \sin (90^{\circ} - \theta) \)
• \( \tan \theta = \cot (90^{\circ} - \theta) \)
• \( \cot \theta = \tan (90^{\circ} - \theta) \)
• Similar relationships exist for secant (sec) and cosecant (csc).

The idea of cofunctions is very useful for simplifying trigonometric expressions and solving equations. By just knowing one function value, you can easily find the corresponding cofunction by using the complementary angle.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is another key element in trigonometry. It represents the ratio of the length of the opposite side to the adjacent side in a right triangle. This can be written as:\[\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\]Some important properties of the tangent function include:

• The function is periodic with a period of \(180^{\circ}\) or \(\pi\) radians.
• It is undefined for angles where the cosine is zero, such as \(90^{\circ}, 270^{\circ}\), etc.
• The tangent value can be positive or negative depending on the quadrant in which the angle is located.

Understanding these properties can help in resolving complex trigonometric problems, such as choosing the correct function based on the specific angle or determining the signs of trigonometric values in different quadrants.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is quite similar to the tangent function but involves a reciprocal relationship. It is defined as the ratio of the adjacent side over the opposite side in a right triangle:\[\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}}\]Some characteristics of the cotangent function include:

• It is undefined when the sine is zero, such as at \(0^{\circ}, 180^{\circ}\), etc.
• Similar to tangent, cotangent has a period of \(180^{\circ}\) or \(\pi\) radians.
• The cotangent is the cofunction of the tangent, meaning \( \cot(90^{\circ} - \theta) = \tan \theta \).

Grasping the concept of cotangent is crucial if you want to excel in trigonometry as it frequently comes up in calculations involving angles and their complementary functions.