Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\cos \frac{7 \pi}{20}\) (b) \(\sin \frac{1}{4}\) (c) \(\tan 1\) (d) \(\csc 0.53\)

Step-by-Step Solution

Answer

(a) \( \sin \frac{3\pi}{20} \), (b) \( \cos \frac{\pi - 1}{4} \), (c) \( \cot(\frac{\pi}{2} - 1) \), (d) \( \sec(\frac{\pi}{2} - 0.53) \).

Step 1: Understanding Cofunctions

Cofunctions relate to the trigonometric functions of complementary angles. For any angle \( \theta \), the basic cofunction identities state that: \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \) and \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \), as well as similar identities for tangent and cosecant. Our task is to express each function as its respective cofunction of a complementary angle.

Step 2: Part (a): Express \(\cos \frac{7\pi}{20}\) as a Cofunction

Using the identity \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \), we can express \( \cos \frac{7\pi}{20} \) as \( \sin(\frac{\pi}{2} - \frac{7\pi}{20}) \). Simplifying the angle, we get \( \frac{\pi}{2} = \frac{10\pi}{20} \), so \( \sin(\frac{10\pi}{20} - \frac{7\pi}{20}) = \sin(\frac{3\pi}{20}) \). Thus, \( \cos \frac{7\pi}{20} = \sin \frac{3\pi}{20} \).

Step 3: Part (b): Express \(\sin \frac{1}{4}\) as a Cofunction

Using the identity \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \), express \( \sin \frac{1}{4} \) as \( \cos(\frac{\pi}{2} - \frac{1}{4}) \). Here, \( \frac{\pi}{2} = \frac{2\pi}{4} \), so \( \cos(\frac{2\pi}{4} - \frac{1}{4}) = \cos(\frac{\pi - 1}{4}) \).

Step 4: Part (c): Express \(\tan 1\) as a Cofunction

Using the identity \( \tan(\theta) = \cot(\frac{\pi}{2} - \theta) \), express \( \tan 1 \) as \( \cot(\frac{\pi}{2} - 1) \). Hence, \( \tan 1 = \cot(\frac{\pi}{2} - 1) \).

Step 5: Part (d): Express \(\csc 0.53\) as a Cofunction

Using the identity \( \csc(\theta) = \sec(\frac{\pi}{2} - \theta) \), express \( \csc 0.53 \) as \( \sec(\frac{\pi}{2} - 0.53) \). Thus, \( \csc 0.53 = \sec(\frac{\pi}{2} - 0.53) \).

Key Concepts

Complementary Angles

Complementary angles are two angles that add up to 90 degrees or \( \frac{\pi}{2} \) radians. This property is crucial in understanding trigonometric cofunctions, as it creates a unique relationship between the angle measures. When one angle is \( \theta \), its complement will be \( \frac{\pi}{2} - \theta \). This relationship forms the basis for cofunction identities in trigonometry.

• Cosine and Sine: If \( \theta \) is an angle, then \( \cos(\theta) = \sin \left( \frac{\pi}{2} - \theta \right) \) and \( \sin(\theta) = \cos \left( \frac{\pi}{2} - \theta \right) \).
• Tangent and Cotangent: These functions are related by the complementary angle as \( \tan(\theta) = \cot \left( \frac{\pi}{2} - \theta \right) \) and \( \cot(\theta) = \tan \left( \frac{\pi}{2} - \theta \right) \).
• Secant and Cosecant: Similar identities exist for \( \csc(\theta) = \sec \left( \frac{\pi}{2} - \theta \right) \) and \( \sec(\theta) = \csc \left( \frac{\pi}{2} - \theta \right) \).

By understanding these relationships, solving trigonometric problems involving complementary angles becomes more intuitive. Complementary angles and their cofunctions play a vital role in simplifications and calculations in both geometry and trigonometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They help simplify expressions and solve trigonometric equations. The cofunction identities make use of complementary angles to relate different trigonometric functions. These identities are essential when rewriting functions in terms of their complements.
For example, using the cofunction identity, we can express \( \sin \theta \) as \( \cos \left( \frac{\pi}{2} - \theta \right) \). This allows one to work interchangeably between sine and cosine, depending on which is more convenient for a given problem. Similarly, \( \tan \theta \) and \( \cot \left( \frac{\pi}{2} - \theta \right) \) offer a way to switch between tangent and cotangent.
These identities aren't just for conversions—they also provide deeper insight into the symmetrical properties of trigonometric functions. Understanding them enhances one's ability to grasp complex concepts such as Fourier transforms or wave functions. They show how interconnected the world of trigonometry is, with each function serving as an essential piece of the larger puzzle.

Circular Functions

Trigonometric functions are often referred to as circular functions because of their geometric definitions on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.

• Sine and Cosine: On the unit circle, \(\sin\theta\) is the y-coordinate and \(\cos\theta\) is the x-coordinate of a point corresponding to an angle \(\theta\).
• Tangent: The tangent function can be interpreted as the slope of the line formed by the radius that corresponds to \(\theta\).

These functions describe how an angle \(\theta\) in radians corresponds to a point on the unit circle. The real beauty of circular functions lies in their periodic nature, which reflects how they repeat values in a predictable pattern. This property is especially valuable when addressing cyclical phenomena in physics and engineering, such as wave behavior or oscillations.
Understanding trigonometric functions in the context of the unit circle not only aids in mastering the functions themselves but also enhances our comprehension of their applications in real-world scenarios. Circular functions extend beyond mere triangles, offering a comprehensive toolset for analyzing angles and their relationships.