Problem 47 - Algebra and Trigonometry

Question

\(47-50\) Find the exact value of the given expression. $$ \sin \left(2 \cos ^{-1} \frac{7}{25}\right) $$

Step-by-Step Solution

Answer

\(\sin \left(2 \cos ^{-1} \frac{7}{25}\right) = \frac{336}{625}\).

Step 1: Understand Cosine Inverse

The expression given involves \( \cos^{-1} \), which is the inverse cosine function. We're given \( \cos^{-1} \left( \frac{7}{25} \right) \). This means we have an angle \( \theta \) such that \( \cos(\theta) = \frac{7}{25} \).

Step 2: Use Trigonometric Identity for Double Angle

We will use the double angle identity for sine: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). To apply this formula, we need both \( \sin(\theta) \) and \( \cos(\theta) \). We already know \( \cos(\theta) = \frac{7}{25} \).

Step 3: Find Sine from Cosine

Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), let's solve for \( \sin(\theta) \). Substitute \( \cos(\theta) = \frac{7}{25} \):\[ \sin^2(\theta) + \left( \frac{7}{25} \right)^2 = 1 \]\[ \sin^2(\theta) + \frac{49}{625} = 1 \]\[ \sin^2(\theta) = 1 - \frac{49}{625} = \frac{576}{625} \]\( \sin(\theta) = \pm \frac{24}{25} \). Since we're dealing with \( \cos^{-1} \) which results in angles from \(0\) to \(\pi\), \( \sin(\theta) \) will be positive, so \( \sin(\theta) = \frac{24}{25} \).

Step 4: Apply the Sine Double Angle Identity

We now have \( \sin(\theta) = \frac{24}{25} \) and \( \cos(\theta) = \frac{7}{25} \). Use the identity: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]Substitute the values:\[ \sin(2\theta) = 2 \times \frac{24}{25} \times \frac{7}{25} \]\[ \sin(2\theta) = \frac{2 \times 24 \times 7}{625} = \frac{336}{625} \].

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine an angle when you know the ratio of sides in a right triangle. They include functions like \( \cos^{-1}, \sin^{-1}, \tan^{-1} \), and more. When you encounter \( \cos^{-1} \left( \frac{7}{25} \right) \), it tells you there's an angle \( \theta \) where the cosine value equals \( \frac{7}{25} \).
Inverse functions are useful because they "reverse" a trigonometric function. For instance, if \( \cos(\theta) = \frac{7}{25} \), then \( \theta = \cos^{-1} \left( \frac{7}{25} \right) \).
Keep in mind:

• \( \cos^{-1}(x) \) will give you an angle in the range from \( 0 \) to \( \pi \).
• The values expressed through inverse trigonometric functions are angles, used to solve further equations.

These functions are vital in situations where you need to resolve angles based on their cosine, sine, or tangent ratios.

Double Angle Formulas

Double angle formulas are crucial for transforming expressions involving double angles into something more manageable. They are derived from the sum formulas of trigonometric functions. A commonly used double angle formula is for sine:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
This formula allows you to calculate the sine of twice an angle using simple sine and cosine values of the original angle, \( \theta \).
Here's how it is applied:

• If \( \sin(\theta) \) and \( \cos(\theta) \) are known, \( \sin(2\theta) \) can be calculated directly.
• In the solution, \( \sin(\theta) = \frac{24}{25} \) and \( \cos(\theta) = \frac{7}{25} \) were used to find \( \sin(2\theta) = \frac{336}{625} \).

This makes it easier to simplify complex trigonometric expressions and solve trigonometric equations more effectively.

Pythagorean Identity

The Pythagorean identity is a trigonometric truth that relates the squares of sine and cosine for any angle within a triangle. It is expressed as:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
This identity stems from the Pythagorean theorem applied to the unit circle. It confirms that for any angle, the sum of the squares of its sine and cosine will always equal one.
In the solution:

• Given \( \cos(\theta) = \frac{7}{25} \), \( \sin(\theta) \) can be found by plugging into the identity.
• Here, the calculation is \( \sin^2(\theta) = 1 - \left( \frac{7}{25} \right)^2 = \frac{576}{625} \), resulting in \( \sin(\theta) = \frac{24}{25} \).

This identity is essential for solving trigonometric problems as it allows you to derive unknown trigonometric values from known ones.