Problem 56 - 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. \(\cos \theta=\frac{1}{2} ; 0^{\circ}<\theta<90^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 2\theta = \frac{\sqrt{3}}{2} \), \( \cos 2\theta = -\frac{1}{2} \), \( \sin \frac{\theta}{2} = \frac{1}{2} \), \( \cos \frac{\theta}{2} = \frac{\sqrt{3}}{2} \).

Step 1: Use the Identity for Sin 2θ

The double angle formula for sine is given by: \[ \sin 2 \theta = 2 \sin \theta \cos \theta \] To find \(\sin 2\theta\), we first need to determine \(\sin \theta\) using the relationship \(\sin^2 \theta + \cos^2 \theta = 1\). Since \(\cos \theta = \frac{1}{2}\), we have: \[ \sin^2 \theta + \left(\frac{1}{2}\right)^2 = 1 \] \[ \sin^2 \theta + \frac{1}{4} = 1 \] Solving gives: \[ \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \] \[ \sin \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \] Now, substitute back into the double angle formula: \[ \sin 2 \theta = 2 \left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2} \]

Step 2: Use the Identity for Cos 2θ

The double angle formula for cosine gives us three different identities:1. \( \cos 2 \theta = \cos^2 \theta - \sin^2 \theta \)2. \( \cos 2 \theta = 2\cos^2 \theta - 1 \)3. \( \cos 2 \theta = 1 - 2\sin^2 \theta \)Using \( \cos^2 \theta - \sin^2 \theta \):\[ \cos 2 \theta = \left(\frac{1}{2}\right)^2 - \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \] Thus, \( \cos 2 \theta = -\frac{1}{2} \).

Step 3: Use the Identity for Sin θ/2

The half-angle formula for sine is given by: \[ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \] Since \(0^{\circ} < \theta < 90^{\circ}\), \(\frac{\theta}{2}\) is also in the first quadrant, so \(\sin \frac{\theta}{2} > 0\).Substitute the value of \(\cos \theta = \frac{1}{2}\): \[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

Step 4: Use the Identity for Cos θ/2

The half-angle formula for cosine is given by: \[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \] Similarly, since \(\frac{\theta}{2}\) is in the first quadrant, \(\cos \frac{\theta}{2} > 0\).Substitute the value of \(\cos \theta = \frac{1}{2}\): \[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{1}{2}}{2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

Key Concepts

Double Angle Formula

The Double Angle Formula is a helpful tool in trigonometry that allows us to find the sine and cosine values of double angles. Specifically, the formula relates a trigonometric function at double the angle to the squares of sine and cosine functions at the original angle. This is especially useful when we know the trigonometric values for a single angle but need them for twice that angle.
For sine, the formula is:

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

To find \( \sin 2 \theta \), we calculated \( \sin \theta \) first, using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
Once \( \sin \theta \) was found to be \( \frac{\sqrt{3}}{2} \), substituting into the double angle formula gave us \( \sin 2 \theta = \frac{\sqrt{3}}{2} \).

For cosine, there are three different identities we can use, but we applied:

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

This calculation used the values \( \cos \theta = \frac{1}{2} \) and \( \sin \theta = \frac{\sqrt{3}}{2} \), leading to \( \cos 2 \theta = -\frac{1}{2} \).
These formulas simplify the process of finding trigonometric values for angles that are multiples of the original.

Half Angle Formula

The Half Angle Formulas are designed to find the sine and cosine of half an angle when we know the trigonometric functions of that original angle. These formulas are particularly useful in simplifying certain integrals and solving equations.
The half angle formulas for sine and cosine are:

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

Since the problem specifies \( 0^{\circ} < \theta < 90^{\circ} \), \( \frac{\theta}{2} \) is within the first quadrant. This means both sine and cosine are positive due to the nature of trigonometric functions in this quadrant.
Substituting \( \cos \theta = \frac{1}{2} \) into the half angle formulas, we found:

• \( \sin \frac{\theta}{2} = \frac{1}{2} \)
• \( \cos \frac{\theta}{2} = \frac{\sqrt{3}}{2} \)

These formulas empower us to determine exact trigonometric values for angles like \( \frac{\theta}{2} \) even when the original angle is easy to manage.

Value of Trigonometric Functions

Understanding the values of trigonometric functions is crucial for solving many math problems. In this exercise, we worked with trigonometric values when given an angle \( \theta \) in the first quadrant, where \( \cos \theta = \frac{1}{2} \).
Using trigonometric identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \), we calculated \( \sin \theta = \frac{\sqrt{3}}{2} \). This identity is fundamental because it assures that for any angle \( \theta \), the squares of the sine and cosine functions sum to one.
With these values, we could apply further identities like the double angle and half angle formulas to find \( \sin 2 \theta \), \( \cos 2 \theta \), \( \sin \frac{\theta}{2} \), and \( \cos \frac{\theta}{2} \).
Knowing how to manipulate these relationships is key to mastering trigonometry. It enables you to tackle more complex mathematical challenges efficiently.

First Quadrant

In trigonometry, the first quadrant refers to the section of the coordinate plane where both x and y coordinates are positive. This quadrant spans angles from 0 to 90 degrees, which corresponds to \( 0 \) to \( \frac{\pi}{2} \) radians.
Angles in this quadrant are particularly significant because:

• Both sine and cosine values are positive.
• The tangent of an angle is positive as well.
• This quadrant serves as the foundational reference for understanding angles in other quadrants.

For the given exercise, knowing that \( \theta \) was in the first quadrant confirmed the positivity of trigonometric functions such as \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \). Keeping the properties of the first quadrant in mind is beneficial in accurately solving trigonometric expressions and equations.