Problem 10 - 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. $$ \sin \theta=\frac{5}{13} ; 90^{\circ}<\theta<180^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{120}{169}\), \(\cos 2\theta = -\frac{49}{169}\), \(\sin \frac{\theta}{2} = \sqrt{\frac{25}{26}}\), \(\cos \frac{\theta}{2} = \frac{1}{\sqrt{26}}\).

Step 1: Determine the quadrant

Given that \(\theta\) is between \(90^{\circ}\) and \(180^{\circ}\), \(\theta\) is in the second quadrant. In this quadrant, \(\sin \theta\) is positive, while \(\cos \theta\) is negative.

Step 2: Use Pythagorean Identity

Since \(\sin \theta = \frac{5}{13}\), we use the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute the known value: \(\left(\frac{5}{13}\right)^2 + \cos^2 \theta = 1\). Calculate \(\frac{25}{169} + \cos^2 \theta = 1\), which simplifies to \(\cos^2 \theta = \frac{144}{169}\). Since \(\theta\) is in the second quadrant, \(\cos \theta = -\frac{12}{13}\).

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

Use the double angle identity \(\sin 2\theta = 2\sin \theta \cos \theta\). Substituting the known values: \(\sin 2\theta = 2 \times \frac{5}{13} \times -\frac{12}{13}\). This simplifies to \(\sin 2\theta = -\frac{120}{169}\).

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

Use the double angle identity \(\cos 2\theta = 2\cos^2 \theta - 1\). Substitute: \(\cos 2\theta = 2 \times \left(-\frac{12}{13}\right)^2 - 1\). This simplifies to \(\cos 2\theta = \frac{120}{169} - 1 = -\frac{49}{169}\).

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

Use the half-angle identity: \(\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}\). Since \(\frac{\theta}{2}\) is between \(45^{\circ}\) and \(90^{\circ}\) (second quadrant where \(\sin\) is positive), it will be positive. Substitute: \(\sin \frac{\theta}{2} = \sqrt{\frac{1 + \frac{12}{13}}{2}}\). Calculate: \(\sin \frac{\theta}{2} = \sqrt{\frac{25}{26}}\).

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

Use the half-angle identity: \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\). Since \(\frac{\theta}{2}\) is between \(45^{\circ}\) and \(90^{\circ}\), it will be positive. Substitute: \(\cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{12}{13}}{2}}\). Calculate: \(\cos \frac{\theta}{2} = \sqrt{\frac{1}{26}}\).

Key Concepts

Double Angle Identities

In trigonometry, double angle identities are incredibly useful for transforming trigonometric expressions that involve angles that are double of another angle, such as transforming \( \sin 2\theta \) and \( \cos 2\theta \). These identities come in quite handy for solving equations and proving other identities.

To find \( \sin 2\theta \), we use the identity: \[ \sin 2\theta = 2\sin \theta \cos \theta \]. This identity multiplies the sine and cosine values of the given angle then scales it by 2. For example, if you know \( \sin \theta = \frac{5}{13} \) and \( \cos \theta = -\frac{12}{13} \) (which has been derived using the Pythagorean Identity), you substitute them into the formula:

• \( \sin 2\theta = 2 \times \frac{5}{13} \times -\frac{12}{13} \)
• This results in \( \sin 2\theta = -\frac{120}{169} \)

Similarly, for \( \cos 2\theta \), the identity is: \[ \cos 2\theta = 2\cos^2 \theta - 1 \]. By substituting the known values, \( \cos 2\theta = \frac{120}{169} - 1 = -\frac{49}{169}\). These expressions show how double angle identities effectively encapsulate the relationships between these trigonometric functions for doubled angles.

Half-Angle Identities

Half-angle identities are another powerful set of formulas used in trigonometry, perfectly suited for finding expressions related to half an angle, such as \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \). These identities allow for calculating trigonometric functions based on the cosine or sine of the full angle.

To find \( \sin \frac{\theta}{2} \), the identity is: \[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \]. The choice of sign depends on the quadrant where the angle is situated. In our given example, since \( \frac{\theta}{2} \) lies in the second quadrant (where sine is positive), we use the positive root:

• Substitute the values: \( \sin \frac{\theta}{2} = \sqrt{\frac{1 + \frac{12}{13}}{2}}\)
• This simplifies to \( \sin \frac{\theta}{2} = \sqrt{\frac{25}{26}} \)

For \( \cos \frac{\theta}{2} \), the identity is: \[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \]. Again, given the second quadrant context of our problem:

• Substitute appropriately: \( \cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{12}{13}}{2}}\)
• This simplifies down to \( \cos \frac{\theta}{2} = \sqrt{\frac{1}{26}} \)

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental and widely used identities in trigonometry. It relates the square of sine and cosine of the same angle with the equation: \[ \sin^2 \theta + \cos^2 \theta = 1 \].

This identity helps connect the sine and cosine functions in such a way that allows for the calculation of one if the other is known. Taking our given condition \( \sin \theta = \frac{5}{13} \), you can calculate \( \cos \theta \) as follows:

• Start with \( \left(\frac{5}{13}\right)^2 + \cos^2 \theta = 1\)
• Simplifying this: \( \frac{25}{169} + \cos^2 \theta = 1 \)
• Solve for the unknown: \( \cos^2 \theta = \frac{144}{169} \)
• Since \( \theta \) is in the second quadrant, where cosine is negative, \( \cos \theta = -\frac{12}{13}\).

It's crucial to remember that the quadrant in which the angle lies determines the sign of \( \cos \theta \), making the Pythagorean Identity not only a computational tool but also a method for understanding trigonometric function behavior across different angle ranges.