Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$1-2 \sin ^{2} 22.5^{\circ}$$

Step-by-Step Solution

Answer

The expression is equal to \( \frac{\sqrt{2}}{2} \).

Step 1: Recall the Double Angle Identity for Cosine

The expression given is similar to the double angle identity for cosine: \ \[ \cos(2\theta) = 1 - 2\sin^2(\theta) \] \ Our goal is to see if the expression can be rewritten as \( \cos(2\theta) \).

Step 2: Match the Expression to the Identity

Recognize that 1-2\( \sin^2(22.5^{\circ}) \) directly fits the identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). Here, \( \theta = 22.5^{\circ} \).

Step 3: Apply the Identity

By identifying \( \theta = 22.5^{\circ} \), we can use the identity: \ \( \cos(2\times 22.5^{\circ}) = 1 - 2\sin^2(22.5^{\circ}) \). Therefore, \ \( 1-2 \sin^2(22.5^{\circ}) = \cos(45^{\circ}) \).

Step 4: Evaluate \( \cos(45^{\circ}) \)

The exact value of \( \cos(45^{\circ}) \) is known from basic trigonometric values. \ Therefore, \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \).

Key Concepts

Double Angle Identity

In trigonometry, the Double Angle Identities are formulas that relate the trigonometric functions of twice an angle to functions of the original angle. These identities help simplify complex trigonometric expressions. For the cosine function, the double angle identity is expressed as:

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

This identity can transform an expression involving \( \sin^2(\theta) \) into an expression involving cosine of a double angle, \( \cos(2\theta) \). This can make calculations easier and provide an exact representation without using decimals.

In the provided problem, the expression \( 1 - 2\sin^2(22.5^{\circ}) \) is immediately recognizable as a form of the double angle identity for cosine. By identifying \( \theta = 22.5^{\circ} \), the expression can be rewritten as \( \cos(2 \times 22.5^{\circ}) = \cos(45^{\circ}) \). This transformation simplifies the expression to a known trigonometric value.

Exact Values

Trigonometric exact values refer to specific angles whose trigonometric function values are known to be precise fractional or irrational numbers, rather than decimals. Many of these angles are commonly encountered like \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \text{and} 90^{\circ} \). They can be determined through geometric considerations, such as examining special triangles, or using identities.

In the context of the exercise, the angle \( 45^{\circ} \) is prominent because the cosine of this angle is a well-known exact value. From the step-by-step solution, we reach \( \cos(45^{\circ}) \), which has an exact value of \( \frac{\sqrt{2}}{2} \). This precision is crucial in mathematics, as it allows calculations to be carried out exactly without the need for approximation, making it easier to understand and apply in further problems.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, alongside sine and tangent. It relates the lengths of the adjacent side and the hypotenuse of a right-angled triangle. The cosine of an angle \( \theta \), denoted \( \cos(\theta) \), tells us about how much of the hypotenuse is in line with the adjacent side as the angle increases from \( 0^{\circ} \) to \( 360^{\circ} \).

For example:

• At \( \theta = 0^{\circ} \), \( \cos(0^{\circ})=1 \), as the entire hypotenuse aligns with the adjacent side.
• At \( \theta = 90^{\circ} \), \( \cos(90^{\circ})=0 \), since the hypotenuse and the adjacent side are perpendicular.
• Similarly, at \( \theta = 45^{\circ}, \cos(45^{\circ})=\frac{\sqrt{2}}{2} \), which means the length is divided equally between the adjacent side and the opposite side.

In this exercise, utilizing the double angle identity allowed us to convert the expression into \( \cos(45^{\circ}) \), which is a commonly used exact value of the cosine function, illustrating how identities make trigonometric problems more manageable.