Problem 27 - Algebra and Trigonometry

Question

Use an identity to find the value of each expression. Do not use a calculator. $$ \sin ^{2} \frac{\pi}{9}+\cos ^{2} \frac{\pi}{9} $$

Step-by-Step Solution

Answer

The value of the expression \( \sin^2 (\frac{\pi}{9}) + \cos^2 (\frac{\pi}{9}) \) is 1.

Step 1: Identify the Trigonometric Identity

Recognize the trigonometric identity for sine and cosine: \( \sin^2 (x) + \cos^2 (x) = 1 \). This identity is valid for any real number x.

Step 2: Apply the Trigonometric Identity

Apply the trigonometric identity \( \sin^2 (x) + \cos^2 (x) = 1 \) to the given expression. Here, \( x = \frac{\pi}{9} \).

Step 3: Substitute and Simplify

Substitute \( \frac{\pi}{9} \) into the trigonometric identity and simplify. We find, \( \sin^2 (\frac{\pi}{9}) + \cos^2 (\frac{\pi}{9}) = 1 \).

Key Concepts

Sine and Cosine Relationship

The relationship between sine and cosine is one of the foundational concepts in trigonometry. At the core of this relationship is the Pythagorean Identity. This identity states that for any angle \( x \), the square of sine of \( x \) plus the square of cosine of \( x \) equals 1. Mathematically, it is expressed as:

• \( \sin^2(x) + \cos^2(x) = 1 \)

This identity is powerful because it holds true regardless of the angle and is the basis for simplifying many trigonometric expressions. It also connects to the Pythagorean theorem in geometry, where \( \sin(x) \) and \( \cos(x) \) can be seen as the lengths of the legs of a right triangle whose hypotenuse is 1.
Understanding this identity allows us to find the values of trigonometric expressions without a calculator by simplifying them using this relationship. Like in the exercise, even without computing the individual values of \( \sin(\frac{\pi}{9}) \) and \( \cos(\frac{\pi}{9}) \), we can conclude that their squares add up to 1.

Unit Circle

The unit circle is a crucial tool in understanding trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The significance of the unit circle in trigonometry lies in its ability to define the sine, cosine, and tangent of an angle.

• Sine of an angle corresponds to the y-coordinate of a point on the unit circle.
• Cosine of an angle corresponds to the x-coordinate of a point on the unit circle.
• The radius of the circle, remaining constant at 1, helps establish the Pythagorean Identity, as \( x^2 + y^2 = 1 \) for any point \((x,y)\) on the unit circle.

Angles are usually measured from the positive x-axis. This representation makes visualization and understanding of trigonometric concepts easier. For instance, the exercise simplifies to 1 due to the inherent geometric properties of the unit circle. Ultimately, the unit circle is a powerful tool for understanding and navigating through trigonometric identities without resorting to calculators.

Trigonometric Simplification

Trigonometric simplification is a valuable skill when working with trigonometric expressions. It involves reducing expressions to simpler forms using known identities and properties.
One of the first steps in simplification is identifying applicable trigonometric identities, like the Pythagorean identity, which directly helped in the given exercise to simplify \[ \sin^2(\frac{\pi}{9}) + \cos^2(\frac{\pi}{9}) = 1 \]This straightforward application of identities indicates how crucial they are in trigonometry.

• Recognizing familiar patterns or forms in an expression can lead to immediate simplification.
• Using identities effectively prevents complex calculations and decreases dependency on calculators.

Practicing simplification enhances problem-solving skills and deepens comprehension of the relationships among various trigonometric functions. Moreover, it allows tackling trigonometric problems with efficiency and confidence.