Problem 31 - Technical Mathematics with Calculus

Question

Cofunctions. Express as a function of the complementary angle. $$\sec 85.6^{\circ}$$

Step-by-Step Solution

Answer

\(\sec 85.6^\circ = \frac{1}{\cos(4.4^\circ)}\)

Step 1: Understanding Cofunctions

Cofunctions are pairs of trigonometric functions that are equal when their angles add up to \(90^\circ\) or \(\frac{\pi}{2}\) radians. The cofunction of secant (sec) is cosine (cos), so we need to find the complement of the given angle, which is \(90^\circ - 85.6^\circ\).

Step 2: Calculate Complementary Angle

Subtract the given angle from \(90^\circ\) to find its complementary angle: \(90^\circ - 85.6^\circ = 4.4^\circ\). This is the angle whose cosine will be the cofunction of \(\sec 85.6^\circ\).

Step 3: Express in Terms of Complementary Angle

Using the fact that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we can express \(\sec 85.6^\circ\) in terms of the complementary angle as \(\sec 85.6^\circ = \frac{1}{\cos(4.4^\circ)}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. There are six primary trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Each function has a specific relationship with a given angle in a right-angled triangle. For instance, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. Similarly, the secant is the reciprocal of the cosine, which means it is the ratio of the length of the hypotenuse to the adjacent side.

Trigonometric functions are widely used not only in geometry and algebra but also in various fields such as engineering, physics, and computer science, as they help solve problems involving angles and distances.

Complementary Angles

Complementary angles are two angles that add up to a right angle, which is exactly 90 degrees or \(\frac{\pi}{2}\) radians. In trigonometry, the concept of complementary angles is crucial because of the cofunction identities that relate the trigonometric functions of an angle to the functions of its complement.

The most common pairs of cofunctions are sine and cosine, tangent and cotangent, and secant and cosecant. For example, the sine of an angle is the same as the cosine of its complementary angle and vice versa. Knowing these relationships is essential when solving problems that ask you to express a trigonometric function in terms of its cofunction.

Secant (sec)

The secant (sec) function is one of the six main trigonometric functions. It is defined as the reciprocal of the cosine function. In a right-angled triangle, for an angle \theta, secant is the ratio of the hypotenuse to the adjacent side, which is represented as \(\frac{1}{\cos(\theta)}\).

While the cosine of an angle gives us a direct ratio, the secant provides a reciprocal relation, which can be especially handy when solving equations or inequalities where the reciprocal of a cosine appears. Moreover, the secant function is important when discussing trigonometric identities and solving complex trigonometric equations.