Problem 33 - 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. $$ \sin \theta=-\frac{1}{4} ; 180^{\circ}<\theta<270^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 2\theta = \frac{\sqrt{15}}{8}, \cos 2\theta = \frac{7}{8}, \sin \frac{\theta}{2} = \sqrt{\frac{4 + \sqrt{15}}{8}}, \cos \frac{\theta}{2} = -\sqrt{\frac{4 - \sqrt{15}}{8}} \)

Step 1: Determine the quadrant of \( \theta \)

Since \( \theta \) is between \( 180^{\circ} \) and \( 270^{\circ} \), \( \theta \) is in the third quadrant. In this quadrant, the sine function is negative, which confirms that \( \sin \theta = -\frac{1}{4} \) is consistent with the given condition.

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

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), solve for \( \cos \theta \).\[\sin^2 \theta = \left(-\frac{1}{4}\right)^2 = \frac{1}{16}\]\[\cos^2 \theta = 1 - \frac{1}{16} = \frac{15}{16}\]Since in the third quadrant, cosine is negative:\[\cos \theta = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4}\]

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

Use the double angle formula for sine:\[\sin 2\theta = 2 \sin \theta \cos \theta\]Substitute the known values:\[\sin 2\theta = 2 \left(-\frac{1}{4}\right) \left(-\frac{\sqrt{15}}{4}\right) = \frac{\sqrt{15}}{8}\]

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

Use the double angle formula for cosine:\[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]Substitute the known values:\[\cos 2\theta = \frac{15}{16} - \frac{1}{16} = \frac{14}{16} = \frac{7}{8}\]

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

Since \( \theta \) is in the third quadrant, \( \frac{\theta}{2} \) will be in the second quadrant where sine is positive.Use the half angle formula for sine:\[\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}\]Substitute the known value for \( \cos \theta \):\[\sin \frac{\theta}{2} = \sqrt{\frac{1 + \frac{\sqrt{15}}{4}}{2}} = \sqrt{\frac{4 + \sqrt{15}}{8}}\]

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

Since \( \frac{\theta}{2} \) is in the second quadrant, \( \cos \frac{\theta}{2} \) will be negative.Use the half angle formula for cosine:\[\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\]Substitute the known value for \( \cos \theta \):\[\cos \frac{\theta}{2} = -\sqrt{\frac{1 - \frac{\sqrt{15}}{4}}{2}} = -\sqrt{\frac{4 - \sqrt{15}}{8}}\]

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions, which hold true for every value of the occurring variables. They serve as a fundamental aspect of trigonometry because they allow us to manipulate equations and prove other trigonometric relationships. One of the core identities is the Pythagorean identity:

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

This identity is paramount because it connects the sine and cosine of the same angle, providing a way to find one trigonometric function if you know the other. For example, in this exercise, since we have \( \sin \theta = -\frac{1}{4} \), we can find \( \cos \theta \) using this identity. Specifically, square the given sine value and subtract it from 1 to find the square of the cosine. This kind of manipulation is essential in solving complex trigonometric equations.

Exploring Half Angle Formulas

Half angle formulas are closely related to double angle formulas and are useful for breaking down trigonometric functions into more manageable components. They are derived from the power reducing formulas and are stated as:

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

In this exercise, we used these formulas to find \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \). First, identify the quadrant which determines the sign of the result. For \( \sin \frac{\theta}{2} \), because \( \frac{\theta}{2} \) is in the second quadrant, sine is positive. Conversely, for \( \cos \frac{\theta}{2} \), it will be negative. These formulas allow us to accurately find the trigonometric function of half the angle, taking care to apply the correct sign based on the appropriate quadrant.

An Insight into Double Angle Formulas

Double angle formulas simplify the computations of trigonometric functions of twice an angle. Here, the sine and cosine double angle formulas are:

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

These formulas are particularly useful in this exercise for calculating \( \sin 2\theta \) and \( \cos 2\theta \). By using the given \( \sin \theta \) and the determined \( \cos \theta \), we calculated \( \sin 2\theta = \frac{\sqrt{15}}{8} \) and \( \cos 2\theta = \frac{7}{8} \). Mastery of these formulas helps in solving trigonometric equations where the angle is doubled, providing a more efficient method than calculating the angle directly.

Understanding the Third Quadrant in Trigonometry

In trigonometry, the third quadrant is the section of the coordinate plane where both sine and cosine functions are negative. This is because the angles in this quadrant range from \(180^{\circ}\) to \(270^{\circ}\), placing them in the negative x and y coordinate spaces. Recognizing which quadrant an angle lies in is crucial, as it directly affects the signs of the trigonometric values:

• In the third quadrant, \( \sin \theta \) is negative.
• \( \cos \theta \) is also negative.

For the given problem, we established that \( \theta \) is in the third quadrant, matching the negative sine value, \( \sin \theta = -\frac{1}{4} \). This guided our solution process, ensuring we chose the correct signs for \( \cos \theta \) and subsequently for the values sourced via half and double angle formulas. Knowing quadrants helps in accurately applying trigonometric principles, especially when determining the sign of solutions in equations.