Problem 1 - 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. $$ \cos \theta=\frac{3}{5} ; 0^{\circ}<\theta<90^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 2\theta = \frac{24}{25} \), \( \cos 2\theta = -\frac{7}{25} \), \( \sin \frac{\theta}{2} = \frac{\sqrt{10}}{5} \), \( \cos \frac{\theta}{2} = \frac{2\sqrt{5}}{5} \).

Step 1: Use Pythagorean identity

Given that \( \cos \theta = \frac{3}{5} \), use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). Substitute the value of \( \cos \theta \) into the identity: \( \sin^2 \theta + \left( \frac{3}{5} \right)^2 = 1 \). This simplifies to \( \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \), so \( \sin \theta = \frac{4}{5} \) considering \( \sin \theta > 0 \) for \( 0^\circ < \theta < 90^\circ \).

Step 2: Calculate \( \sin 2\theta \) using double angle identity

Use the formula for \( \sin 2\theta \), which is \( 2 \sin \theta \cos \theta \). Substitute the known values: \( \sin 2\theta = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25} \).

Step 3: Calculate \( \cos 2\theta \) using double angle identity

Use the double angle identity for cosine \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Substitute the known values: \( \cos 2\theta = \left( \frac{3}{5} \right)^2 - \left( \frac{4}{5} \right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \).

Step 4: Calculate \( \sin \frac{\theta}{2} \) using half angle identity

Use the half angle identity \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \). Substitute the known value for \( \cos \theta \): \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{2}{5}} = \frac{\sqrt{10}}{5} \).

Step 5: Calculate \( \cos \frac{\theta}{2} \) using half angle identity

Use the half angle identity \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). Substitute the known value for \( \cos \theta \): \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the sum of the square of sine and cosine equals 1. Mathematically, this is represented as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is immensely useful because it allows us to find one trigonometric function when we know the other. For example, if you know \( \cos \theta \), you can find \( \sin \theta \) by rearranging the formula:

• Subtract \( \cos^2 \theta \) from both sides: \( \sin^2 \theta = 1 - \cos^2 \theta \).
• Take the square root: \( \sin \theta = \sqrt{1 - \cos^2 \theta} \).

In our exercise, \( \cos \theta \) was given as \( \frac{3}{5} \). Using the Pythagorean identity, we calculated \( \sin \theta \) to be \( \frac{4}{5} \). This calculation is crucial for solving many trigonometric problems, including those involving other identities used in this exercise.

Double Angle Identity

The Double Angle Identity is another powerful tool in trigonometry. It helps simplify expressions involving angles that are double the size of a given angle, such as \( 2\theta \). The double angle identities for sine and cosine are:

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

These formulas are particularly useful for analyzing angles like \( 2\theta \) where you know \( \sin \theta \) and \( \cos \theta \). In our example:

• We used \( \sin 2\theta = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25} \).
• For \( \cos 2\theta \), we used \( \cos 2\theta = (\frac{3}{5})^2 - (\frac{4}{5})^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \).

This identity helps transform products of sine and cosine into a single cosine or sine function at twice the angle, simplifying many problems.

Half Angle Identity

The Half Angle Identity is a key concept to find the sine or cosine of half an angle. These identities are especially valuable when working with angles that are fractions, such as \( \frac{\theta}{2} \). The half angle identities for sine and cosine are:

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

In our specific exercise, we calculated:

• \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \frac{\sqrt{10}}{5} \).
• \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \frac{2\sqrt{5}}{5} \).

Understanding these identities can significantly simplify the process of working with angles not directly available on most standard trigonometric tables.