Problem 3 - Algebra 2

Question

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \sin \theta=\frac{1}{2} ; 0^{\circ}<\theta<90^{\circ} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Given and Required Values

We're given that \( \sin \theta = \frac{1}{2} \) and \( 0^{\circ} < \theta < 90^{\circ} \). We need to find \( \sin 2\theta \), \( \cos 2\theta \), \( \sin \frac{\theta}{2} \), and \( \cos \frac{\theta}{2} \).

Step 2: Finding \( \theta \)

Since \( \sin \theta = \frac{1}{2} \), and knowing that for angles in the first quadrant \( 0^{\circ} < \theta < 90^{\circ} \), it implies \( \theta = 30^{\circ} \) because \( \sin 30^{\circ} = \frac{1}{2} \).

Step 3: Calculate \( \sin 2\theta \)

We use the double angle formula for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Knowing \( \theta = 30^{\circ} \), \( \cos \theta = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \). Therefore, \( \sin 2\theta = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \).

Step 4: Calculate \( \cos 2\theta \)

Use the double angle formula for cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \sin \theta = \frac{1}{2} \). So, \( \cos 2\theta = \left( \frac{\sqrt{3}}{2} \right)^2 - \left( \frac{1}{2} \right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \).

Step 5: Calculate \( \sin \frac{\theta}{2} \)

Use the half angle formula for sine: \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \). So \( \sin \frac{30^{\circ}}{2} = \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \). Simplifying, \( \sin 15^{\circ} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \).

Step 6: Calculate \( \cos \frac{\theta}{2} \)

Use the half angle formula for cosine: \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). For \( \cos 15^{\circ} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \). Simplifying, \( \cos 15^{\circ} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \).

Key Concepts

Double Angle Formulas

Double angle formulas are special trigonometric identities that help us calculate the sine and cosine of twice an angle. These formulas allow us to find the trigonometric functions of an angle that is double another angle, hence the name. There are two primary double angle identities:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

These formulas are powerful tools, simplifying the process of finding these values without needing to determine the angle itself directly. Take the example of finding \( \sin 2\theta \) when \( \sin \theta = \frac{1}{2} \) and \( \theta = 30^{\circ} \). We simply need to compute \( \cos \theta = \frac{\sqrt{3}}{2} \), then plug these into the formula:\[ \sin 2\theta = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]For \( \cos 2\theta \), using \( \cos^2 \theta = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \) and \( \sin^2 \theta = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \), we get:\[ \cos 2\theta = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \]

Half Angle Formulas

Half angle formulas are used to determine the sine and cosine of a half angle. These identities are especially useful when working with smaller angles or angles not as easily memorized. The fundamental half angle identities are as follows:

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

By using these formulas, you can derive values like \( \sin \frac{\theta}{2} \) or \( \cos \frac{\theta}{2} \) from known sine and cosine of \( \theta \). For example, to find \( \sin 15^{\circ} \) and \( \cos 15^{\circ} \) given \( \theta = 30^{\circ} \), we first need \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \). Substitute into the half-angle formulas:\[ \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \]\[ \cos 15^{\circ} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \]These calculations show how powerful half-angle formulas are for breaking down and understanding complex angles.

Sine and Cosine Values

Knowing specific values for sine and cosine of common angles is incredibly beneficial when dealing with trigonometric identities. The functions \( \sin \theta \) and \( \cos \theta \) take on standard values at special angles, often referred to as reference angles, such as \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \( 90^{\circ} \). These angles help to simplify computations in trigonometry.For example, at \( \theta = 30^{\circ} \), we know:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)

These values help us quickly determine other trigonometric values through identities such as the double angle and half angle formulas. Additionally, these known values are useful in solving equations, transformations, and analyzing periodic functions, making them cornerstones in the study of trigonometry.