Cracking the Code: A Step-by-Step Explanation of (sin x)² + (cos x)² = 1

In the world of mathematics, certain equations and formulas are like hidden codes waiting to be cracked. One such formula that often puzzles students is the trigonometric identity (sin x)² + (cos x)² = 1. At first glance, this equation may seem complex and confusing, but with a step-by-step explanation, you can unravel its secrets and understand the beauty behind it.

Understanding Trigonometric Identities

Before we delve into the specifics of (sin x)² + (cos x)² = 1, let’s first establish a basic understanding of trigonometric identities. In trigonometry, identities are equations that are true for all values of the variable within their domains. These identities often involve trigonometric functions such as sine, cosine, and tangent.

The Pythagorean Identity

One of the fundamental trigonometric identities is the Pythagorean identity, which states that for any angle x, the equation sin²(x) + cos²(x) = 1 holds true. This identity is derived from the Pythagorean theorem and forms the basis for many other trigonometric identities.

Deriving (sin x)² + (cos x)² = 1

Now, let’s take a closer look at how we can derive the identity (sin x)² + (cos x)² = 1 using the Pythagorean identity.

Step 1: Start with the Pythagorean Identity

We begin by starting with the Pythagorean identity sin²(x) + cos²(x) = 1

Step 2: Square both Sides

Next, we square both sides of the equation to get rid of the square roots. This gives us (sin²(x) + cos²(x))² = 1

Step 3: Expand the Squares

Expanding the squares on the left side of the equation, we get sin⁴(x) + 2sin²(x)cos²(x) + cos⁴(x) = 1

Step 4: Simplify the Equation

By replacing sin²(x) with 1 – cos²(x) and cos²(x) with 1 – sin²(x), we simplify the equation to obtain:

1 – cos²(x) + 2(1 – cos²(x))(1 – sin²(x)) – 1 + sin²(x) = 1

Step 5: Further Simplification

Expanding and simplifying the equation further, we eventually arrive at the identity (sin x)² + (cos x)² = 1

FAQs

What is the significance of the trigonometric identity (sin x)² + (cos x)² = 1?

The identity (sin x)² + (cos x)² = 1 is a fundamental trigonometric relationship that is used extensively in calculus, geometry, and physics. It allows us to relate the sine and cosine functions of an angle x to each other in a simple and elegant way.

How can I use (sin x)² + (cos x)² = 1 in practical applications?

In practical applications, the identity (sin x)² + (cos x)² = 1 is often used to simplify complex trigonometric expressions, prove other trigonometric identities, and solve trigonometric equations. It serves as a cornerstone for understanding the relationships between sine and cosine functions.

Are there other important trigonometric identities I should be aware of?

Yes, there are several other important trigonometric identities, such as the sum and difference identities, double-angle identities, and half-angle identities. These identities play key roles in various branches of mathematics and are essential for solving a wide range of problems.

Can you provide another example of how to derive a trigonometric identity?

Sure! Let’s take a look at the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). By applying the sum of angles formula for sine, sin(x + x), and simplifying the expression, we can derive the double-angle identity.

Conclusion

In conclusion, the trigonometric identity (sin x)² + (cos x)² = 1 is a beautiful and powerful equation that highlights the relationship between sine and cosine functions. By following the step-by-step explanation outlined above, you can gain a deeper understanding of this fundamental identity and its significance in the world of mathematics. So, next time you encounter this equation, remember that it’s not just a code to be cracked, but a key to unlocking the mysteries of trigonometry.