Problem 18 - Calculus Single Variable

Question

\(\theta\) is a number between 0 and \(\pi / 2\). Calculate the unevaluated trigonometric function from the given information. \(\sin (\theta / 2) ; \cos (\theta)=3 / 7\)

Step-by-Step Solution

Answer

\(\sin(\theta/2) = \sqrt{\frac{2}{7}}\)

Step 1: Identify the Relationship

Given that \(\cos(\theta) = \frac{3}{7}\), we want to calculate \(\sin(\theta/2)\). Use the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\) to find \(\sin(\theta)\).

Step 2: Calculate \(\sin(\theta)\)

Use the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\):\[ \sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{3}{7}\right)^2} = \sqrt{1 - \frac{9}{49}} = \sqrt{\frac{40}{49}} = \frac{\sqrt{40}}{7} = \frac{2\sqrt{10}}{7} \]

Step 3: Utilize the Half-Angle Identity

Use the half-angle identity \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \) to find \(\sin(\theta/2)\).

Step 4: Calculate \(\sin(\theta/2)\)

Plug \(\cos(\theta) = \frac{3}{7}\) into the half-angle formula:\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{3}{7}}{2}} = \sqrt{\frac{\frac{7}{7} - \frac{3}{7}}{2}} = \sqrt{\frac{\frac{4}{7}}{2}} = \sqrt{\frac{4}{14}} = \sqrt{\frac{2}{7}} \]

Key Concepts

Understanding Half-Angle Identities

Half-angle identities are extremely useful, especially when dealing with complex trigonometric expressions. They help in evaluating trigonometric values of angles that are not directly accessible from common angle measures. Take, for instance, if you know the cosine of an angle, you can readily find the sine or cosine of half that angle using the half-angle identities.

For sine, the half-angle identity is given by:

• \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \)

This formula allows you to break down the problem into a simpler form and work with known values. It is essential to note the square root when applying this identity, as it will affect whether your result is positive or negative. Remember, the choice of sign depends on the quadrant in which the half-angle lies.

Using the example given: if \( \cos(\theta) = \frac{3}{7} \), substituting this into the half-angle formula simplifies the process of finding \( \sin(\theta/2) \). This showcases the power of half-angle identities in simplifying trigonometric calculations.

Exploring Pythagorean Identities

Pythagorean identities are foundational tools used in trigonometry to relate the squares of the sine, cosine, and tangent functions. The primary Pythagorean identity states:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \)

This means that knowing one of the function values allows you to find the other. It's a direct extension of the Pythagorean theorem since if you visualize a right triangle, the cosine and sine represent the adjacent and opposite sides, while the hypotenuse is 1 in the unit circle model.

In our example, we started with \( \cos(\theta) = \frac{3}{7} \). By applying the Pythagorean identity, we solved for \( \sin(\theta) \):

\[ \sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{3}{7}\right)^2} \]

This calculation allowed us to find \( \sin(\theta) = \frac{2\sqrt{10}}{7} \), showing how this identity is paramount in connecting various trigonometric expressions. Always be mindful that these identities help in converting between different trigonometric forms that can otherwise be hard to compute.

Trigonometric Functions and Their Importance

Trigonometric functions, namely sine, cosine, and tangent, comprise the core concepts of trigonometry. Each function is based on ratios in a right triangle relative to given angles, primarily when working within the unit circle.

• \( \sin(\theta) \) is 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.

These functions are essential because they provide tools to model periodic behaviors like sound waves, light waves, and seasonal changes. In mathematics, they serve as the backbone for dictating relationships between angles and sides in triangles.

In the exercise at hand, knowing \( \cos(\theta) \) and computing \( \sin(\theta/2) \) exemplifies how trigonometric functions simplify problems involving half angles. By applying trigonometric identities meticulously, one can derive values that would otherwise be complicated to handle directly. Mastering these functions and their identities opens up fields of problem-solving in calculus and beyond, advancing comprehension of both theoretical and practical mathematics.