Question

Cofunctions. Express as a function of the complementary angle. $$\tan 19^{\circ}$$

Short answer

\(\tan(19^{\circ}) = \cot(71^{\circ})\)

Step-by-step solution

Understand Cofunctions

Cofunctions are pairs of trigonometric functions that are equal when their angles sum to 90 degrees (a right angle). The tangent function's cofunction is the cotangent function, which means for an angle \(\theta\), \(\tan(\theta)\) is equivalent to \(\cot(90^{\circ}-\theta)\).

Find the Complementary Angle

The complementary angle of \(19^{\circ}\) is found by subtracting it from \(90^{\circ}\) because complementary angles sum up to \(90^{\circ}\). So, the complementary angle of \(19^{\circ}\) is \(90^{\circ} - 19^{\circ} = 71^{\circ}\).

Express as a Function of the Complementary Angle

Using the cofunction relationship between tangent and cotangent, express \(\tan(19^{\circ})\) as a function of \(71^{\circ}\), which is the complementary angle. Therefore \(\tan(19^{\circ})\) becomes \(\cot(71^{\circ})\).

Key concepts

Complementary Angles

Complementary angles play a crucial role in trigonometry, forming the foundation for understanding cofunctions. Two angles are considered complementary when the sum of their measures equals 90 degrees. This is a fundamental concept to grasp because it tells us that each angle essentially completes the other to form a right angle.

For instance, if you have an angle that measures 30 degrees, its complementary angle would measure 60 degrees because combined, they must reach that magic number - 90 degrees. This concept becomes particularly useful when we deal with trigonometric functions because it establishes a unique relationship between them, such as between sine and cosine, or as seen in our example, between tangent and cotangent.

Tangent and Cotangent Relationship

In trigonometry, each function has a cofunction with which it shares a very special relationship, especially when dealing with complementary angles. For tangent and cotangent, this relationship is straightforward but vital for solving certain types of problems.

The tangent of an angle is the cotangent of its complementary angle. In simpler terms, if you have an angle \( \theta \), then \( \tan(\theta) = \cot(90^\circ - \theta) \). This reciprocal connection means that if you know the value of the tangent, you can easily find the cotangent for the angle that complements it to 90 degrees, and vice versa. When we talk about \( \tan(19^\circ) \), we are implicitly referring to the cotangent of its complementary angle, which is \( \cot(71^\circ) \).

Trigonometric Functions

The pillars of trigonometry are its functions, which include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions help us understand and calculate the relationships between the angles and sides of triangles, especially right triangles. Each function provides a different perspective on the angle, with tangent, for example, representing the ratio of the opposite side to the adjacent side of a triangle.

These functions are not isolated; they are interconnected through various identities and relationships, such as the Pythagorean identity or the cofunction identities we have explored. Mastering these connections enables you to transition smoothly between functions and simplify complex trigonometric expressions, which is essential for solving a wide array of problems in both pure and applied mathematics.