Problem 15 - 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. $$ \cos \theta=-\frac{1}{4} ; 90^{\circ}<\theta<180^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{\sqrt{15}}{8}\), \(\cos 2\theta = -\frac{7}{8}\), \(\sin \frac{\theta}{2} = \frac{\sqrt{10}}{4}\), \(\cos \frac{\theta}{2} = \frac{\sqrt{6}}{4}\).

Step 1: Determine sin(θ) using Pythagorean identity

Given that \( \cos \theta = -\frac{1}{4} \) and \( \sin^2 \theta + \cos^2 \theta = 1 \), solve for \( \sin \theta \).\[ \sin^2 \theta = 1 - \left(-\frac{1}{4}\right)^2 = 1 - \frac{1}{16} = \frac{15}{16} \] Since \( 90^\circ < \theta < 180^\circ \), \( \sin \theta \) is positive, thus: \[ \sin \theta = \frac{\sqrt{15}}{4} \]

Step 2: Calculate sin(2θ) and cos(2θ)

Use the double angle identities to find \( \sin 2\theta \) and \( \cos 2\theta \):- \( \sin 2\theta = 2 \sin \theta \cos \theta \)\[ \sin 2\theta = 2 \left(\frac{\sqrt{15}}{4}\right)\left(-\frac{1}{4}\right) = -\frac{\sqrt{15}}{8} \]- \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)\[ \cos 2\theta = \left(-\frac{1}{4}\right)^2 - \left(\frac{\sqrt{15}}{4}\right)^2 = \frac{1}{16} - \frac{15}{16} = -\frac{14}{16} = -\frac{7}{8} \]

Step 3: Calculate sin(θ/2) and cos(θ/2)

Use the half-angle identities:- \( \sin\frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \)\[ \sin\frac{\theta}{2} = \sqrt{\frac{1 + \frac{1}{4}}{2}} = \sqrt{\frac{5}{8}} = \frac{\sqrt{10}}{4} \]- \( \cos\frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \)\[ \cos\frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{4}}{2}} = \sqrt{\frac{3}{8}} = \frac{\sqrt{6}}{4} \]

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the most fundamental identities in trigonometry, and it is essential for solving various trigonometric problems. This identity is expressed as:\[\sin^2 \theta + \cos^2 \theta = 1\]This equation reveals the relationship between the sine and cosine of an angle \(\theta\). It can be derived from the Pythagorean theorem in a right-angled triangle. Remember, this identity holds true for all angles.Key Points:

• Use this identity to find the missing sine or cosine value if one of them is known.
• It helps determine the possible sign (positive or negative) of the trigonometric functions based on the quadrant in which the angle lies.
• For example, when \( \cos \theta = -\frac{1}{4} \), applying the identity helps find \( \sin \theta \).

Knowing this identity and how to manipulate it can make solving trigonometric equations much easier.

double angle identities

The double angle identities involve expressions for trigonometric functions where angles are doubled, such as \(\sin 2\theta\) and \(\cos 2\theta\). These identities are crucial for simplifying and solving equations.Double Angle Formulas:

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

In solving for \( \sin 2\theta \) and \( \cos 2\theta \), use these formulas to substitute the known values of \( \sin \theta \) and \( \cos \theta \). This is especially helpful when given that \( \cos \theta = -\frac{1}{4} \) and the task is to find these double angle values.Tips:

• These identities help in converting problems involving single angles into double angles, making it possible to solve complex trigonometric equations.
• They are particularly useful in integration and other advanced calculus applications.

Understanding these identities will allow you to tackle a wide range of trigonometric problems with greater efficiency.

half-angle identities

Half-angle identities are powerful tools in trigonometry, primarily used to find the sine and cosine of half an angle, such as \(\sin \frac{\theta}{2}\) and \(\cos \frac{\theta}{2}\). These identities can simplify expressions and solve equations involving multiples or fractions of angles.Half-Angle Formulas:

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

These formulas use the cosine of the entire angle to find the sine and cosine of half the angle. In the context of this exercise, using \( \cos \theta = -\frac{1}{4} \) helps to compute \(\sin \frac{\theta}{2}\) and \(\cos \frac{\theta}{2}\).Benefits:

• Half-angle identities are particularly useful when angles are halved in integration problems or when evaluating specific trigonometric expressions.
• They provide exact values, which is useful in geometric and analytical contexts.

Understanding these identities can greatly enhance your ability to solve trigonometric problems efficiently and accurately.