Problem 35 - 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. $$ \sin \theta=\frac{2}{5} ; 90^{\circ}<\theta<180^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{4\sqrt{21}}{25}\), \(\cos 2\theta = \frac{17}{25}\), \(\sin \frac{\theta}{2} = \sqrt{\frac{1 + \frac{\sqrt{21}}{5}}{2}}\), \(\cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{\sqrt{21}}{5}}{2}}\).

Step 1: Identify Quadrant and Determine Cosine

Since \(90^{\circ} < \theta < 180^{\circ}\), \(\theta\) is in the second quadrant where sine is positive and cosine is negative. Use \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\). Calculate:\[\cos \theta = -\sqrt{1 - \sin^2 \theta} = -\sqrt{1 - \left(\frac{2}{5}\right)^2} = -\sqrt{1 - \frac{4}{25}} = -\sqrt{\frac{21}{25}} = -\frac{\sqrt{21}}{5}\]

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

Use the double angle identity \(\sin 2\theta = 2 \sin \theta \cos \theta\):\[\sin 2\theta = 2 \cdot \frac{2}{5} \cdot -\frac{\sqrt{21}}{5} = -\frac{4\sqrt{21}}{25}\]

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

Use the double angle identity \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\):\[\cos 2\theta = \left(-\frac{\sqrt{21}}{5}\right)^2 - \left(\frac{2}{5}\right)^2 = \frac{21}{25} - \frac{4}{25} = \frac{17}{25}\]

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

Use the half-angle identity \(\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}\):\[\sin \frac{\theta}{2} = \sqrt{\frac{1 + \frac{\sqrt{21}}{5}}{2}}\]\( heta\) being in the second quadrant implies \(\frac{\theta}{2}\) is actually in the first quadrant, ensuring result is positive.

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

Use the half-angle identity \(\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}}\):\[\cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{\sqrt{21}}{5}}{2}}\]Since \(\frac{\theta}{2}\) is in the first quadrant, the cosine value is also positive.

Key Concepts

Double Angle Identities

Double angle identities are essential in trigonometry because they help simplify various trigonometric functions, allowing easier calculation and understanding of angles and expressions. These identities incorporate angles that are double a given angle, such as \(2\theta\).

In trigonometry, the double angle identities for sine and cosine are:

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

To use these effectively, you need to know the values of \(\sin\) and \(\cos\) for the initial angle \(\theta\). For example, given \(\sin \theta = \frac{2}{5}\) and knowing \(\theta\) is in the second quadrant, you can deduce that \(\cos \theta = -\frac{\sqrt{21}}{5}\) due to the identity \(\sin^2 \theta + \cos^2 \theta = 1\).

Once you've established both sine and cosine of the angle, you plug these into the double angle identities. This will derive exact values for \(\sin 2\theta\) and \(\cos 2\theta\), thereby providing insights into the behavior of trigonometric functions at double the initial angle.

Half Angle Identities

Half angle identities are vital for solving trigonometry problems that involve finding the value of trigonometric functions at an angle divided by two. They bridge the link when dealing with angles that are halved, such as \(\frac{\theta}{2}\). This can help find precise values for sine and cosine of the half angle.

The basic identities for half angles are:

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

These identities are particularly useful when the original angle \(\theta\) is known from which \(\cos \theta\) can be computed. Using these identities helps in determining the exact value for the sine and cosine of angles like \(\frac{\theta}{2}\).

For instance, if \(\theta\) is in the second quadrant (where \(90^\circ < \theta < 180^\circ\)), then \(\frac{\theta}{2}\) falls into the first quadrant. This change of quadrant is crucial because sine and cosine values are positive in the first quadrant, simplifying their computation directly.

Trigonometric Functions

Trigonometric functions are foundational elements in mathematics, especially in the field of geometry and trigonometry. They describe relationships between the angles and sides of triangles, and they extend these relationships to general angles beyond triangles such as in circles and waves.

Key trigonometric functions include:

• The sine function \(\sin \theta\), which represents the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function \(\cos \theta\), which is the ratio of the adjacent side to the hypotenuse in a right triangle.
• The tangent function \(\tan \theta\), defined as the quotient of \(\sin \theta\) over \(\cos \theta\).

Understanding these functions allows students to solve problems involving triangles, model periodic phenomena, and navigate complex calculations involving angles and measurements in various disciplines. Mastery of using identities such as double angle and half angle enriches the skill set for problem-solving, offering methods to simplify expressions and compute angles effectively.