Problem 19 - Calculus Single Variable

Question

\(\theta\) is a number between 0 and \(\pi / 2\). Calculate the unevaluated trigonometric function from the given information. \(\cos (\theta / 2) ; \sin (\theta)=5 / 13\)

Step-by-Step Solution

Answer

\( \cos(\theta/2) = \frac{5\sqrt{26}}{26} \)

Step 1: Understand the Given Information

We are given the value of \( \sin \theta = \frac{5}{13} \) and need to calculate \( \cos(\theta/2) \). The angle \( \theta \) is between 0 and \( \pi/2 \), meaning it's in the first quadrant where all trigonometric ratios are positive.

Step 2: Use the Pythagorean Identity

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substituting \( \sin \theta = \frac{5}{13} \), we get: \[\left(\frac{5}{13}\right)^2 + \cos^2 \theta = 1\] \[\frac{25}{169} + \cos^2 \theta = 1\] \[\cos^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}\]Taking the square root, \( \cos \theta = \frac{12}{13} \) because \( \theta \) is in the first quadrant.

Step 3: Apply the Half-Angle Formula

The half-angle formula for cosine is: \[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]Substitute \( \cos \theta = \frac{12}{13} \):\[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \frac{12}{13}}{2}} = \pm \sqrt{\frac{25}{26}} \]Since \( \theta / 2 \) is also in the first quadrant, \( \cos(\theta/2) \) is positive.

Step 4: Simplify the Expression

Calculate the positive square root:\[ \cos \left( \frac{\theta}{2} \right) = \sqrt{\frac{25}{26}} = \frac{5}{\sqrt{26}} \]Rationalizing the denominator, we get:\[ \cos \left( \frac{\theta}{2} \right) = \frac{5\sqrt{26}}{26} \]

Key Concepts

Half-Angle Formula

The half-angle formula is a handy tool in trigonometry that allows us to find the cosine of half an angle if we know the cosine of the whole angle. This formula is especially useful when evaluating trigonometric functions for angles that are not standard ones, like 30°, 45°, etc. The formula for the cosine of half an angle states: \[ \cos\left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \] Depending on the quadrant where the angle falls, the sign of the square root changes. - In the first quadrant where all basic trigonometric functions are positive, we use the positive sign. - For other quadrants, the sign reflects whether cosine is positive or negative there. In our exercise, since \( \theta/2 \) is also in the first quadrant, we confidently select the positive square root.

Pythagorean Identity

The Pythagorean Identity is central in trigonometry and provides a direct link between the sine and cosine of the same angle. It is expressed as: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This equation confirms that the sum of the squares of sine and cosine equals one for any angle \( \theta \). Given one of these functions, we can always find the other. For example, in our problem, we were given \( \sin \theta = \frac{5}{13} \). By rearranging the identity: - First, substitute the given sine value: \( \left( \frac{5}{13} \right)^2 + \cos^2 \theta = 1 \) - Simplifying gives \( \cos^2 \theta = \frac{144}{169} \) - Finally, taking the square root and choosing the positive value (since \( \theta \) is in the first quadrant) yields \( \cos \theta = \frac{12}{13} \). This calculation lays the groundwork to apply further formulas, such as the half-angle formula.

First Quadrant Trigonometry

Trigonometry in the first quadrant is particularly approachable because all trigonometric functions take on positive values here. The first quadrant is where the angle \( \theta \) is between 0 and \( \frac{\pi}{2} \) radians (or 0 and 90 degrees). In this region: - \( \sin \theta \) and \( \cos \theta \) are both positive. - This positive nature simplifies problems involving square roots or selecting signs of trigonometric expressions. - For example, with the half-angle formula, since \( \theta/2 \) is also in the first quadrant, we confidently choose the positive square root. Understanding these quadrantal properties helps solve problems easily by knowing the sign of trigonometric functions without additional calculations.