Problem 28 - Advanced Engineering Mathematics

Question

Prove that \(\cos ^{2} z+\sin ^{2} z=1\)

Step-by-Step Solution

Answer

The identity \( \cos^2 z + \sin^2 z = 1 \) is based on the equation of the unit circle.

Step 1: Recall the Unit Circle Definition

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. For any angle \( z \), the coordinates of any point on the unit circle are given by \( (\cos z, \sin z) \).

Step 2: Apply Pythagorean Theorem to the Unit Circle

According to the definition of the unit circle, the equation for a circle with radius 1 centered at the origin is \( x^2 + y^2 = 1 \). For any point on the unit circle, substitute \( x = \cos z \) and \( y = \sin z \), yielding \( (\cos z)^2 + (\sin z)^2 = 1 \).

Step 3: Simplify the Expression

Simplify \( (\cos z)^2 + (\sin z)^2 = 1 \) as \( \cos^2 z + \sin^2 z = 1 \), matching the given identity and confirming it holds true.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( z \), the sum of the squares of the sine and cosine of \( z \) is always equal to 1. Mathematically, this is expressed as \( \cos^2 z + \sin^2 z = 1 \). To understand why this identity holds, we can trace it back to the Pythagorean Theorem, which relates the sides of a right triangle: \( a^2 + b^2 = c^2 \). In the context of the unit circle, the hypotenuse \( c \) is 1, as the radius of the unit circle is 1.

• Simplifies complex trigonometric expressions by reducing them.
• Verifies solutions of trigonometric equations.
• Provides a basis for more complex identities and theorems.

This identity not only simplifies calculations but also aids in graphing and transforming problems within trigonometry.

Unit Circle

Understanding the unit circle is key to mastering trigonometry. The unit circle is a powerful tool because it allows us to visualize angles and their corresponding trigonometric values.

• Centered at the origin of the coordinate system (0,0).
• Has a radius of exactly 1, making mathematical calculations simpler.
• Every point on the circumference represents an angle \( z \), given by coordinates \((\cos z, \sin z)\).

Using the unit circle, we can derive various trigonometric identities, including the Pythagorean Identity. As it simplifies the computation of trigonometric functions, it is invaluable in solving trigonometric equations and understanding periodic functions.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships between the angles and sides of triangles. Although it originated from the study of triangles, its principles are applicable in many fields, from physics to engineering.

• Focuses on functions such as sine, cosine, and tangent.
• Essential for understanding waves, oscillations, and circles.
• Allows for the solving of real-world problems, such as calculating heights and distances indirectly.

Trigonometry leverages concepts like the unit circle to explore periodicity, waves, and oscillations, offering powerful ways to model and analyze cyclical phenomena. With these foundations, students can solve complex problems involving angles and dimensions in both two and three dimensions.