Problem 17 - Calculus Single Variable

Question

\(\theta\) is a number between 0 and \(\pi / 2\). Calculate the unevaluated trigonometric function from the given information. \(\sin (2 \theta) ; \cos (\theta)=4 / 5\)

Step-by-Step Solution

Answer

\( \sin(2\theta) = \frac{24}{25} \)

Step 1: Understanding the Relationship between the Functions

We are given that \( \cos(\theta) = \frac{4}{5} \) and asked to calculate \( \sin(2\theta) \). The relationship between sine and cosine for this exercise is that \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). To use this formula, we need to find \( \sin(\theta) \).

Step 2: Calculate \( \sin(\theta) \) using the Pythagorean Identity

We use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Given \( \cos(\theta) = \frac{4}{5} \), calculate \( \sin(\theta) \) by rewriting the identity: \( \sin^2(\theta) = 1 - \left(\frac{4}{5}\right)^2 \).

Step 3: Compute \( \sin^2(\theta) \)

Substitute the given value into the identity: \( \sin^2(\theta) = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \).

Step 4: Solve for \( \sin(\theta) \)

To find \( \sin(\theta) \), take the square root of \( \sin^2(\theta) \): \( \sin(\theta) = \sqrt{\frac{9}{25}} = \frac{3}{5} \). Since \( \theta \) is between 0 and \( \pi/2 \), \( \sin(\theta) \) is positive.

Step 5: Calculate \( \sin(2\theta) \) using Double Angle Formula

Now that we know both \( \sin(\theta) = \frac{3}{5} \) and \( \cos(\theta) = \frac{4}{5} \), use the double angle formula: \( \sin(2\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25} \).

Key Concepts

Double Angle Formula

One powerful tool in trigonometry is the double angle formula. This formula helps us find the sine, cosine, or tangent of twice a given angle using the trigonometric functions of the original angle. For instance, if you know the sine and cosine of an angle \( \theta \), you can determine \( \sin(2\theta) \). The formula for sine is particularly interesting and is expressed as: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \] This indicates that to find \( \sin(2\theta) \), you need both the sine and cosine of the angle \( \theta \). This double angle formula is derived from the basic sine addition formulas, yet it's extremely useful in simplifying calculations, especially when dealing with trigonometric equations.

• It can simplify complex expressions and make solving problems easier.
• It has applications in physics, engineering, and computer graphics where trigonometric values of angles are often needed.

In the context of the double angle formula, given \( \cos(\theta) = \frac{4}{5} \) and after finding \( \sin(\theta) \), we use this formula to calculate \( \sin(2\theta) \). This process showcases the importance and applicability of double angle formulas in solving practical trigonometric problems.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It establishes a relationship between a sine and cosine for any angle \( \theta \). Written as: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This identity is derived from the Pythagorean theorem and is immensely useful in trigonometry to find missing values of sine or cosine, given the other. You can rearrange the formula to solve for either function if you have information about one:

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

In practical problems, like the one given where \( \cos(\theta) = \frac{4}{5} \), substituting this value into the Pythagorean identity allows us to directly find \( \sin(\theta) \). Therefore, the Pythagorean identity isn't just a formula; it's a gateway to solving problems where direct measurement or calculation of the trigonometric functions isn't straightforward. By understanding how to manipulate this identity, you can tackle a wide variety of trigonometric problems with confidence.

Sine and Cosine Relationship

The sine and cosine functions are core components of trigonometry. They describe the relationship between the sides of a right triangle relative to its angles, and they are also the building blocks for more complex trigonometric identities and formulas.

• \( \sin(\theta) \) represents the ratio of the opposite side to the hypotenuse.
• \( \cos(\theta) \) represents the ratio of the adjacent side to the hypotenuse.

These functions are not only fundamental to understanding basic triangle properties but also to working with angles in different units, such as radians. When given \( \cos(\theta) = \frac{4}{5} \), knowing the sine-cosine relationship helps find \( \sin(\theta) \). This relationship is key when converting and combining angles using identities. The known values of sine and cosine are often used with other functions to derive more complex trigonometric expressions, such as tangents or secants. Additionally, understanding their relationship forms the basis for understanding graphs of these functions. This knowledge is essential in fields such as physics for wave motion or engineering for signal processing.