Problem 20 - Precalculus Essentials

Question

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$1-2 \sin ^{2} \frac{\pi}{12}$$

Step-by-Step Solution

Answer

The expression \(1-2 \sin^2 \frac{\pi}{12}\) is equivalent to \(\cos(\frac{\pi}{6})\) and evaluates to \(\frac{\sqrt{3}}{2}\).

Step 1: Apply the double-angle formula for cosine

First, apply the double-angle formula for cosine to rewrite the given expression. Recognize that given expression is a form of double-angle formula, \(1 - 2 \sin^2 \theta = \cos (2\theta)\). This gives: \(\cos (2 \times \frac{\pi}{12}) = \cos (\frac{\pi}{6})\)

Step 2: Evaluate the cosine of the double angle

Next, calculate the value of \(\cos(\frac{\pi}{6})\). If we refer to the unit circle or a right triangle in the first quadrant that has an angle \(\frac{\pi}{6}\), we can find out that \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They are essential tools in simplifying expressions and solving trigonometric equations. These identities allow us to express complex trigonometric relationships in simpler forms. One commonly used identity is the double angle formula for cosine, which states:\[ \cos(2\theta) = 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1 \]This identity helps us convert an expression involving a squared sine or cosine term into a cosine of a double angle. By recognizing the identity in an expression, like in our problem, we can easily rewrite it and further calculate its value.

• Allows simplification of trigonometric expressions.
• Helps in solving complex equations.
• Provides a means to evaluate angles in simpler forms.

Understanding these identities is crucial for making connections between different trigonometric functions and using them effectively in various mathematical contexts.

Cosine Function

The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is often abbreviated as \( \cos \). For every angle \( \theta \), the cosine function provides the horizontal coordinate of the point on the unit circle corresponding to that angle.The cosine function is periodic, meaning it repeats its values in regular intervals, specifically every \(2\pi\) radians. This property is useful in many applications, including solving trigonometric equations and signal analysis. In our exercise, the cosine of \(\frac{\pi}{6}\) was evaluated using known exact values on the unit circle.

• Used to find coordinates of angles on the unit circle.
• Periodicity is \(2\pi\), repeating every full circle.
• Essential in understanding wave patterns and oscillations.

grasping how to use the cosine function effectively enables solving a wide variety of problems related to angles and triangles.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions is a fundamental skill in trigonometry. The exact values often correspond to the angles that are commonly used in mathematics, such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), and so on. For these specific angles, the trigonometric functions have well-known exact values that can be derived from special triangles or the unit circle.For instance, the exact value of \( \cos(\frac{\pi}{6}) \) is \( \frac{\sqrt{3}}{2} \). Recognizing and memorizing these values is immensely helpful for quickly solving problems without the need for a calculator.

• Rudimentary for quick problem-solving.
• Derived from geometric interpretations of trigonometric functions.
• Memory aids in efficiently tackling exercises.

Having these values at your disposal simplifies calculations and enriches understanding of how trigonometric principles are applied in practice.