Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

Step-by-Step Solution

Answer

The identity \((\sin x + \cos x)^2 = 1 + \sin 2x\) is true for all values of x.

Step 1: Expand the square on the left-hand side

First, square the expression \(\sin x + \cos x\). This will give \((\sin x+\cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x\).

Step 2: Apply the Pythagorean Identity

Now apply the Pythagorean identity which is \(\sin^2(x) + \cos^2(x) = 1\). Replacing \(\sin^2 x + \cos^2 x\) in the expression with 1 we get \(1 + 2 \sin x \cos x\).

Step 3: Apply the Double-Angle Identity

The double-angle identity of sine, \(\sin 2x = 2 \sin x \cos x\) is comparable to the expression received after step 2, \(1 + 2 \sin x \cos x\). Replacing the term \(2 \sin x \cos x\) with \(\sin 2x\) gives us the expression \(1 + \sin 2x\).

Step 4: Confirm the Identity

Now that we have worked down the expression to \(1 + \sin 2x\), we can compare it to the original right-hand side of the equation, which is exactly the same. Therefore we have shown that the original identity is true algebraically.

Key Concepts

Pythagorean identity

The Pythagorean identity is a cornerstone of trigonometry. It states:

• \(\sin^2(x) + \cos^2(x) = 1\)

This identity stems from the Pythagorean theorem, applied to a unit circle where the hypotenuse is 1.
In our exercise, we used this identity in step 2 to simplify \((\sin x + \cos x)^2\).
After expanding, \(\sin^2 x + \cos^2 x\) was replaced with 1. This conversion is crucial because it reduces complexity, making advanced trigonometric identities easier to handle.
The identity helps express trigonometric equations in simpler terms, aiding in verification processes or algebraic manipulations.
Remember, the Pythagorean identity works small miracles by binding two primary trigonometric functions into unity.

double-angle identity

The double-angle identity is a mathematical shortcut to simplify expressions involving trigonometric functions at double the angle, like \(2x\).
For sine, it states:

• \(\sin 2x = 2 \sin x \cos x\)

In our step-by-step solution, we used the double-angle identity to transform \(2 \sin x \cos x\) into \(\sin 2x\).
This transformation is key for simplifying expressions with terms that mirror double the basic angle composition.
Recognizing terms as double-angle equivalents can significantly simplify algebraic verification of trigonometric identities.
These identities are not only functional in proving other equations but also pivotal in calculus and beyond, where trigonometric substitutions streamline complex problems.

trigonometric functions

Trigonometric functions like sine \(\sin x\) and cosine \(\cos x\) are foundational in mathematics. They are defined based on ratios in a right triangle or points on the unit circle.
These functions have several important properties and identities, such as period and amplitude, that allow them to model periodic phenomena.
In our exercise, leveraging their inherent relationships through identities allowed us to prove a complex equation algebraically.

• Sine (\(\sin x\)): Vertical coordinate of a point on the unit circle.
• Cosine (\(\cos x\)): Horizontal coordinate of a point on the unit circle.

These functions' periodic nature makes them indispensable in physics, engineering, and even biology. Understanding their behavior, combined with identities, allows us to solve intricate mathematical problems with elegance.