Problem 6 - Algebra 2

Question

Find the exact value of each expression by using the half-angle formulas. \(\cos \frac{19 \pi}{12}\)

Step-by-Step Solution

Answer

The exact value of \( \cos \frac{19 \pi}{12} \) is \(-\frac{\sqrt{2 - \sqrt{3}}}{2}\).

Step 1: Identify Angle to Use

The given angle is \( \frac{19 \pi}{12} \). To use the half-angle formula, we first need to express this angle in terms of a known angle. Notice that \( \frac{19 \pi}{12} = \frac{1}{2} \times \frac{19 \pi}{6} \). Therefore, we can write \( \theta = \frac{19 \pi}{6} \), and our task is to evaluate \( \cos \frac{\theta}{2} \).

Step 2: Use Relation with Known Angles

We need to find a known angle such that when halved gives us \( \frac{19 \pi}{12} \). Recognize that \( \frac{19 \pi}{12} \) is derived by halving \( \frac{1}{2} \times \frac{19 \pi}{6} \), and we can rewrite \( \frac{19 \pi}{6} \) as a sum \( 2\pi + \frac{7 \pi}{6} \). Thus \( \frac{19 \pi}{12} = \frac{1}{2} \left(2\pi + \frac{7 \pi}{6}\right) \).

Step 3: Half-Angle Formula for Cosine

The half-angle formula for cosine is \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \). Thus, we need to find \( \cos \left(\pi + \frac{7\pi}{6}\right) \). Since \( \pi + \frac{7\pi}{6} \) is the angle for which we calculate \( \cos \theta \), and this is equivalent to \( \frac{19\pi}{6} \), where the cosine of angle \( \theta = \frac{7\pi}{6} \) is half of a known angle family.

Step 4: Find Cosine of \( \theta \) for Half-Angle

We know \( \cos \pi \) and \( \cos \left(\frac{7\pi}{6}\right) \). Since \( \frac{7\pi}{6} = \pi + \frac{\pi}{6} \), \( \cos \left(\pi + \frac{\pi}{6}\right) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2} \) since cosine is negative in the third quadrant.

Step 5: Apply and Simplify the Formula

Substitute \( \cos \theta = -\frac{\sqrt{3}}{2} \) into our half-angle formula: \( \cos \frac{19\pi}{12} = \pm \sqrt{\frac{1 + (-\frac{\sqrt{3}}{2})}{2}} \). Simplify: \( = \pm \sqrt{\frac{2 - \sqrt{3}}{4}} \). Since \( \frac{19\pi}{12} \) is an angle in the second quadrant, cosine is negative, thus \( \cos \frac{19\pi}{12} = -\sqrt{\frac{2 - \sqrt{3}}{4}} = -\frac{\sqrt{2 - \sqrt{3}}}{2} \).

Key Concepts

Half-Angle Formulas

The half-angle formulas in trigonometry help you find the sine, cosine, and tangent of angles that are half the size of known angles. Understanding these formulas is important as they allow you to simplify complex trigonometric expressions effectively.
Let's focus on the cosine half-angle formula. It is normally expressed as:

• \[\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\]

The important thing to note here is the \( \pm \) sign, which indicates that the result could be positive or negative depending on which quadrant the angle \( \frac{\theta}{2} \) resides. Quadrants refer to the division of the coordinate system, which we'll discuss further.
In our exercise, the angle \( \frac{19 \pi}{12} \) was expressed in terms of \( \theta = \frac{19 \pi}{6} \), which simplified the problem into using this half-angle formula for computation. It is essential to first find \( \cos \theta \) and then apply it to the half-angle formula for determining \( \cos \frac{\theta}{2} \).

Cosine Function

The cosine function is one of the primary trigonometric functions. It is valuable for determining the x-coordinate of a point on a unit circle at a given angle. In general, the cosine of an angle \( \theta \) is the adjacent side over the hypotenuse in a right-angled triangle.
For our exercise, you need to find the cosine of angles derived from \( \theta = \frac{19\pi}{6} \). To find \( \cos \theta \), we look at the angle \( \pi + \frac{\pi}{6} \). Since this angle is in the third quadrant, we use the identity:

• \[\cos(\pi + \alpha) = -\cos \alpha\]

Applying this to our angle, we find that:

• \[\cos \left( \pi + \frac{\pi}{6} \right) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}\]

Thus, the cosine value at this angle guides us in computing the half-angle value using the half-angle formula explained before.

Trigonometric Quadrants

Trigonometric quadrants are key when determining the sign of the trigonometric functions. The coordinate plane is divided into four quadrants, with different sign conventions for each trigonometric function.
Each quadrant denotes a portion of the coordinate system:

• Quadrant I: All trigonometric functions (sine, cosine, tangent) are positive.
• Quadrant II: Only sine is positive.
• Quadrant III: Only tangent is positive.
• Quadrant IV: Only cosine is positive.

In the context of our problem, the angle \( \frac{19\pi}{12} \) is in the second quadrant. Therefore, cosine should be negative since it is the only positive function in the fourth quadrant. Remembering the properties of these quadrants helped determine the sign of \( \cos \frac{19\pi}{12} \) as negative, finalizing the solution to \(-\frac{\sqrt{2 - \sqrt{3}}}{2}\). This understanding ensures the correct application of the half-angle formula with an accurate reflection of the trigonometric sign conventions.