Problem 30 - 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. $$ \cos \theta=\frac{1}{6} ; 0^{\circ}<\theta<90^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 2\theta = \frac{\sqrt{35}}{18}, \cos 2\theta = -\frac{17}{18}, \sin \frac{\theta}{2} = \frac{\sqrt{15}}{6}, \cos \frac{\theta}{2} = \frac{\sqrt{21}}{6} \).

Step 1: Determine \( \sin \theta \)

We know \( \cos \theta = \frac{1}{6} \). Since \( 0^{\circ}<\theta<90^{\circ} \), \( \sin \theta \) can be found using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). \( \sin^2 \theta = 1 - \left( \frac{1}{6} \right)^2 = 1 - \frac{1}{36} = \frac{35}{36} \). Therefore, \( \sin \theta = \sqrt{\frac{35}{36}} = \frac{\sqrt{35}}{6} \).

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

Use the double-angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substitute the values from previous steps, \( \sin 2\theta = 2 \left( \frac{\sqrt{35}}{6} \right) \left( \frac{1}{6} \right) = \frac{2\sqrt{35}}{36} = \frac{\sqrt{35}}{18} \).

Step 3: Compute \( \cos 2\theta \)

Use the double-angle identity for cosine: \( \cos 2\theta = 2 \cos^2 \theta - 1 \). Substitute \( \cos \theta = \frac{1}{6} \): \( \cos 2\theta = 2 \left( \frac{1}{6} \right)^2 - 1 = \frac{2}{36} - 1 = \frac{1}{18} - 1 = -\frac{17}{18} \).

Step 4: Calculate \( \sin \frac{\theta}{2} \)

For \( \sin \frac{\theta}{2} \), use the formula \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \). Substitute \( \cos \theta = \frac{1}{6} \): \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{6}}{2}} = \sqrt{\frac{\frac{5}{6}}{2}} = \sqrt{\frac{5}{12}} = \frac{\sqrt{15}}{6} \).

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

For \( \cos \frac{\theta}{2} \), use the formula \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). Substitute \( \cos \theta = \frac{1}{6} \): \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{1}{6}}{2}} = \sqrt{\frac{\frac{7}{6}}{2}} = \sqrt{\frac{7}{12}} = \frac{\sqrt{21}}{6} \).

Key Concepts

Double-Angle Identities

Double-angle identities are formulas used to express trigonometric functions of double angles, like \(2\theta\), in terms of single angles \(\theta\). These identities can simplify calculations and are extremely helpful in trigonometry.
One key identity to know for sine is:

• \( \sin 2\theta = 2\sin\theta\cos\theta \)

This formula tells us that the sine of double an angle is twice the product of the sine and cosine of the original angle.
This relationship is handy for solving problems where you know \(\sin\theta\) and \(\cos\theta\) and need to calculate \(\sin 2\theta\).
Another important identity is for cosine:

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

This identity is particularly useful because it allows you to find \(\cos 2\theta\) using only \(\cos\theta\), without even needing to calculate \(\sin\theta\).
Such formulas help in simplifying the trigonometric expressions and are essential for solving complex trigonometric equations.

Half-Angle Identities

Half-angle identities are another set of useful trigonometric formulas. They help calculate trigonometric functions for half of an angle, such as \(\frac{\theta}{2}\), using known values of the angle \(\theta\). These identities are very helpful in calculus and geometry.
Here are the half-angle identities for sine and cosine:

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

The sine half-angle identity allows you to find \(\sin \frac{\theta}{2}\) using just the cosine of the original angle.
This can be particularly useful when \(\cos \theta\) is more easily available, or when it would simplify your solution.
Similarly, the cosine half-angle identity lets you calculate \(\cos \frac{\theta}{2}\) from \(\cos \theta\), making it a powerful tool in trigonometry.
These formulas also determine the sign of the result based on the quadrant in which the half-angle \(\frac{\theta}{2}\) lies, ensuring accurate solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angle values. They are fundamental tools that enable simplification and solution of trigonometric equations and expressions. Here are some crucial identities to remember:

• Pythagorean Identity: \(\sin^2\theta + \cos^2\theta = 1\)

This foundational identity relates sine and cosine, asserting their squared sums equal 1. This is a go-to identity for any trigonometric problem involving \(\sin\theta\) and \(\cos\theta\).

• Reciprocal Identities such as \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), and \(\cot\theta = \frac{1}{\tan\theta}\).

These relationships show how the primary trigonometric functions relate to their reciprocals, aiding in expressing trigonometric functions in various forms.
Using these identities properly can simplify complex trigonometric expressions and make solving equations easier, a critical skill in mathematics.