Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\sin \frac{\pi}{12}\) (b) \(\cos 0.64\) (c) \(\tan \sqrt{2}\) (d) \(\sec 1.2\)

Step-by-Step Solution

Answer

(a) \( \cos \frac{5\pi}{12} \), (b) \( \sin(\frac{\pi}{2} - 0.64) \), (c) \( \cot(\frac{\pi}{2} - \sqrt{2}) \), (d) \( \csc(\frac{\pi}{2} - 1.2) \).

Step 1: Understanding Complementary Angles

Complementary angles are two angles that add up to 90 degrees or \( \frac{\pi}{2} \) radians. The concept of cofunctions involves trigonometric functions of complementary angles.

Step 2: The Cofunction Identity

The cofunction identities relate trigonometric functions to their complements: \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \), \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \), \( \tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right) \), and \( \sec(\theta) = \csc\left(\frac{\pi}{2} - \theta\right) \).

Step 3: Applying Cofunction Identities for (a)

For \( \sin \frac{\pi}{12} \), use the identity: \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \). Substitute \( \theta = \frac{\pi}{12} \) to find \( \cos\left(\frac{\pi}{2} - \frac{\pi}{12}\right) = \cos\frac{5\pi}{12} \).

Step 4: Applying Cofunction Identities for (b)

For \( \cos 0.64 \), use the identity: \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \). Substitute \( \theta = 0.64 \) to find \( \sin\left(\frac{\pi}{2} - 0.64\right) \).

Step 5: Applying Cofunction Identities for (c)

For \( \tan \sqrt{2} \), use the identity: \( \tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right) \). Substitute \( \theta = \sqrt{2} \) to find \( \cot\left(\frac{\pi}{2} - \sqrt{2}\right) \).

Step 6: Applying Cofunction Identities for (d)

For \( \sec 1.2 \), use the identity: \( \sec(\theta) = \csc\left(\frac{\pi}{2} - \theta\right) \). Substitute \( \theta = 1.2 \) to find \( \csc\left(\frac{\pi}{2} - 1.2\right) \).

Key Concepts

Complementary Angles

Complementary angles are two angles whose sum is exactly 90 degrees or \( \frac{\pi}{2} \) radians. They are important in trigonometry because they help us understand relationships between different trigonometric functions. For example, if you have one angle, you can find its complement by subtracting from 90 degrees or \( \frac{\pi}{2} \) radians. This concept is widely used to relate the sine, cosine, tangent, and other trigonometric functions of angles to those of their complementary angles.

In geometry, you often deal with right triangles, where the two non-right angles are complementary. This means that if one of these angles is known, the other can be easily determined. Complementary angles are crucial when learning about cofunctions, as the trigonometric identity bridges these functions with their complements.

Cofunction Identities

Cofunction identities are special relationships between trigonometric functions for complementary angles. These identities tell us how one trigonometric function relates to another through complementary angles. Here are the main cofunction identities:

• \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \)
• \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \)
• \( \tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right) \)
• \( \sec(\theta) = \csc\left(\frac{\pi}{2} - \theta\right) \)
• \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \)
• \( \cot(\theta) = \tan\left(\frac{\pi}{2} - \theta\right) \)

These identities are derived from the fact that the sine and cosine, tangent and cotangent, secant and cosecant are cofunctions of each other. Understanding these identities allows for simplifying expressions and solving equations that involve trigonometric functions with complementary angles. They are handy when you want to express one function in terms of its cofunction to solve trigonometric equations.

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics used to relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are based on the ratios of sides in a right triangle.

• \( \sin(\theta) \) is defined as the ratio of the opposite side to the hypotenuse.
• \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse.
• \( \tan(\theta) \) is the ratio of the opposite side to the adjacent side.

Other functions like secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)) are reciprocals of the primary functions. These functions extend beyond right triangles and are used in various applications, including calculus, physics, engineering, and architecture. They are essential in understanding waves, oscillations, and many other phenomena. In the context of this exercise, understanding trigonometric functions and their identities helps students convert expressions into their complementary functions for easier interpretation and solution.