Problem 55 - 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. \(\sin \theta=\frac{3}{5} ; 0^{\circ}<\theta<90^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{24}{25}, \cos 2\theta = \frac{7}{25}, \sin \frac{\theta}{2} = \frac{\sqrt{10}}{10}, \cos \frac{\theta}{2} = \frac{3\sqrt{10}}{10}\)

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

Given \(\sin \theta = \frac{3}{5}\) and knowing that \(0^{\circ} < \theta < 90^{\circ}\), we use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\).Substitute \(\sin \theta\): \(\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1\)Calculate: \(\frac{9}{25} + \cos^2 \theta = 1\)\(\cos^2 \theta = 1 - \frac{9}{25}\)\(\cos^2 \theta = \frac{16}{25}\)Taking the square root gives:\(\cos \theta = \frac{4}{5}\) since \(0^{\circ} < \theta < 90^{\circ}\) and \(\cos \theta\) is positive.

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

Use the double-angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).Substitute the known values:\(\sin 2\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5}\)Calculate:\(\sin 2\theta = 2 \cdot \frac{12}{25}\)\(\sin 2\theta = \frac{24}{25}\).

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

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

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

Use the half-angle formula for sine: \(\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}\).Substitute \(\cos \theta = \frac{4}{5}\):\(\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{4}{5}}{2}}\)Compute:\(\sin \frac{\theta}{2} = \sqrt{\frac{\frac{1}{5}}{2}}\)\(\sin \frac{\theta}{2} = \sqrt{\frac{1}{10}}\)\(\sin \frac{\theta}{2} = \frac{\sqrt{10}}{10}\).

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

Use the half-angle formula for cosine: \(\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}}\).Substitute \(\cos \theta = \frac{4}{5}\):\(\cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{4}{5}}{2}}\)Compute:\(\cos \frac{\theta}{2} = \sqrt{\frac{\frac{9}{5}}{2}}\)\(\cos \frac{\theta}{2} = \sqrt{\frac{9}{10}}\)\(\cos \frac{\theta}{2} = \frac{3\sqrt{10}}{10}\).

Key Concepts

Double-Angle Formulas

Double-angle formulas are useful tools in trigonometry that allow you to express trigonometric functions of double angles, such as \(2\theta\), in terms of single angles, \(\theta\). These formulas are particularly handy for simplifying expressions involving trigonometric functions and solving equations.
The double-angle formulas for sine and cosine are:

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

To find \(\sin 2\theta\) and \(\cos 2\theta\) using these formulas, you first need the values of \(\sin \theta\) and \(\cos \theta\). Once you have these, it is straightforward to substitute them into the formulas to find the double angles.
In the exercise, \(\sin 2\theta\) was calculated as \(\frac{24}{25}\) and \(\cos 2\theta\) as \(\frac{7}{25}\), which shows how efficiently these formulas can be applied.

Half-Angle Formulas

Half-angle formulas are another set of trigonometric identities that help to express the sine and cosine of half-angles like \(\frac{\theta}{2}\). These formulas are particularly useful when you need precise values of these functions for half-angles, especially when working with angles not easily calculated otherwise. The formulas are:

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

To use these half-angle identities, you must know the value of \(\cos \theta\). Once obtained, you substitute into the formulas to find the half-angle sines and cosines.
From the exercise, \(\sin \frac{\theta}{2}\) was calculated as \(\frac{\sqrt{10}}{10}\) and \(\cos \frac{\theta}{2}\) as \(\frac{3\sqrt{10}}{10}\), illustrating their application. This precision is key when solving trigonometric equations that involve half-angles.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental trigonometric identities, expressing the relationship between the sine and cosine of an angle. It is derived from the Pythagorean theorem, hence the name. The identity states:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity allows us to find one trigonometric ratio if the other is given. For example, if you know \(\sin \theta\), you can easily find \(\cos \theta\) using:\[\cos^2 \theta = 1 - \sin^2 \theta\]In the original exercise, the Pythagorean Identity was used to determine \(\cos \theta\) as well. Given \(\sin \theta = \frac{3}{5}\), we calculated \(\cos \theta = \frac{4}{5}\).
Understanding this identity is crucial as it forms the basis for many other trigonometric formulas and is often used in trigonometric problem solving.

Trigonometric Ratios

Trigonometric ratios, namely sine, cosine, and tangent, are the ratios of the sides of a right triangle regarding its angles. They are essential for defining the trigonometric functions and solving various problems in trigonometry.
For a given angle \(\theta\):

• The sine ratio is \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• The cosine ratio is \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• The tangent ratio is \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)

In right triangles, these ratios can be used to solve for unknown sides or angles. Knowing one ratio helps to find the others using identities like the Pythagorean Identity. For instance, in the exercise, knowing \(\sin \theta = \frac{3}{5}\) helped to find \(\cos \theta\) using the Pythagorean Identity.
Trigonometric ratios serve as the groundwork not only for the identities but also for understanding the behavior of the functions for different angles.