Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\csc \theta=4$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{1}{4} \) and \( \cos \theta = \frac{\sqrt{15}}{4} \).

Step 1: Understand the Given Information

We are given that \( \csc \theta = 4 \). The csc (cosecant) is the reciprocal of the sine function. Therefore, \( \sin \theta = \frac{1}{\csc \theta} = \frac{1}{4} \).

Step 2: Use the Pythagorean Identity

The Pythagorean Identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). We know \( \sin \theta = \frac{1}{4} \), so substitute this into the identity: \( \left(\frac{1}{4}\right)^2 + \cos^2 \theta = 1 \).

Step 3: Solve for \( \cos \theta \)

Calculate \( \left(\frac{1}{4}\right)^2 = \frac{1}{16} \). Therefore, \( \frac{1}{16} + \cos^2 \theta = 1 \). Solve for \( \cos^2 \theta \) by subtracting \( \frac{1}{16} \) from both sides: \( \cos^2 \theta = 1 - \frac{1}{16} = \frac{15}{16} \).

Step 4: Determine \( \cos \theta \)

To find \( \cos \theta \), take the square root of \( \cos^2 \theta \): \( \cos \theta = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4} \). Since \( \theta \) is an acute angle, the cosine is positive.

Key Concepts

Cosecant

Cosecant, often abbreviated as \( \csc \), is one of the six fundamental trigonometric functions. It is particularly important when dealing with angles and triangles. The cosecant of an angle \( \theta \), is defined as the reciprocal of the sine of \( \theta \). This means:

• If \( \csc \theta = 4 \), then \( \sin \theta = \frac{1}{4} \)
• This relationship is crucial to solving problems where the sine of an angle is not readily available

Knowing the \( \csc \) function simplifies the complexity of certain trigonometric equations and helps in the transition between known and unknown values.

Moreover, understanding cosecant assists in analyzing the behavior of angles and their respective sine values relative to the unit circle. It is especially useful in various applications, like calculus and physics, where the properties of wave functions are investigated.

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry. It states that for any angle \( \theta \):\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is defined based on the famous Pythagorean theorem and applies to all angles.

When applied, as in the provided example, it helps find other trigonometric values if one is known:

• Using \( \sin \theta = \frac{1}{4} \), substituting into the identity yields:
• \( \left(\frac{1}{4}\right)^2 + \cos^2 \theta = 1 \)
• This can be further simplified to find \( \cos \theta \)

The identity becomes particularly valuable in confirming relations between trigonometric functions, ensuring accuracy particularly in more complex applications.

Acute Angle

An acute angle is any angle that is less than 90 degrees. These angles are common in various geometric figures and real-world applications such as construction and engineering. In trigonometry:

• Trigonometric functions for acute angles such as sine, cosine, and tangent are always positive
• This property is especially useful when interpreting angles in the first quadrant of the unit circle

For the given problem, recognizing that \( \theta \) is an acute angle allows us to deduce that \( \cos \theta \) is positive due to the properties of trigonometric functions in the first quadrant.

Understanding the nature of acute angles aids in simplifying solutions and ensuring applications align with geometric rules. This basic yet profound concept is a foundational element in trigonometry and helps maintain consistency across different mathematical tasks.