Problem 58 - Algebra 2

Question

Find the exact value of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. \(\sin \theta=\frac{4}{5} ; 0^{\circ}<\theta<90^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{24}{25}, \cos 2\theta = -\frac{7}{25}, \sin \frac{\theta}{2} = \frac{1}{\sqrt{5}}, \cos \frac{\theta}{2} = \frac{2}{\sqrt{5}}\).

Step 1: Recall the Trigonometric Identity for Cosine

Since \( \theta \) is in the first quadrant, both sine and cosine are positive. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta = \frac{4}{5} \) into the identity: \( \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \). This simplifies to \( \frac{16}{25} + \cos^2 \theta = 1 \).

Step 2: Solve for \(\cos \theta\)

Subtract \( \frac{16}{25} \) from 1 to find \( \cos^2 \theta \). This gives \( \cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \). Taking the square root gives \( \cos \theta = \frac{3}{5} \), since \( \theta \) is in the first quadrant and cosine is positive.

Step 3: Use the Double Angle Formulas

Use the formulas \( \sin 2\theta = 2 \sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Substitute \( \sin \theta = \frac{4}{5} \) and \( \cos \theta = \frac{3}{5} \) into the expressions. \( \sin 2\theta = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25} \) and \( \cos 2\theta = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \).

Step 4: Use the Half-Angle Formulas

The formulas are \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \) and \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). Substitute \( \cos \theta = \frac{3}{5} \) into these. So, \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{\frac{2}{5}}{2}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \) and \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \).

Key Concepts

Double Angle Formulas

The double angle formulas are important identities in trigonometry. They express the trigonometric functions of double angles, like \(2\theta\), in terms of single angle functions. These formulas are particularly useful in simplifying expressions and solving equations.

**Key Double Angle Formulas:**
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]

By knowing the sine and cosine of an angle \(\theta\), you can easily find \(\sin 2\theta\) and \(\cos 2\theta\). For instance, if you know \(\sin \theta = \frac{4}{5}\) and \(\cos \theta = \frac{3}{5}\), you can compute:

• \(\sin 2\theta = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25}\)
• \(\cos 2\theta = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = -\frac{7}{25}\)

Using these formulas, you can solve problems involving double angles quickly and precisely.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry connecting the sine and cosine of an angle. It is expressed as:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

This identity is incredibly useful for finding one trigonometric function when the other is known. For instance, given \(\sin \theta = \frac{4}{5}\), you can determine \(\cos \theta\).

• Substitute \(\sin \theta = \frac{4}{5}\) into the identity:
  
  \( \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \)
• Simplify:
  
  \( \frac{16}{25} + \cos^2 \theta = 1 \)
• Solve for \(\cos^2 \theta\) by subtracting:
  
  \( \cos^2 \theta = \frac{9}{25} \)

Since \(\theta\) is in the first quadrant, where cosine is positive, take the positive square root: \(\cos \theta = \frac{3}{5}\). This identity is foundational for solving many problems involving trigonometric functions.

Half-Angle Formulas

Half-angle formulas allow us to find the sine and cosine of half an angle, \(\frac{\theta}{2}\), using the cosine of the angle \(\theta\). These formulas are crucial when the usual trigonometric values are difficult to find directly.

**Primary Half-Angle Formulas:**

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

With \(\cos \theta = \frac{3}{5}\), you find:

• \(\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \frac{1}{\sqrt{5}}\)
• \(\cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \frac{2}{\sqrt{5}}\)

These formulas are especially useful for detailed angle calculations and transformations in trigonometry.

First Quadrant Angles

In trigonometry, angles in the first quadrant, that is, between \(0^\circ\) and \(90^\circ\), have both sine and cosine values that are positive. This quadrant is often the easiest to work with because the values for all standard trigonometric functions are straightforward.

**Characteristics of First Quadrant Angles:**

• \(\sin \theta\) and \(\cos \theta\) are positive.
• Useful for simplifying calculations since it involves basic positive projections.
• Helps to confirm that certain derived values like square roots will be positive.

For example, if \(\sin \theta = \frac{4}{5}\), in the first quadrant, the associated cosine \(\cos \theta\) will be positive, confirmed by solving the Pythagorean identity: \(\cos \theta = \frac{3}{5}\). Recognizing this quadrant can simplify both understanding and calculations of trigonometric functions.