Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$3+\sin ^{2} z=4-\cos ^{2} z$$

Step-by-Step Solution

Answer

The identity \(3+\sin ^{2} z=4-\cos ^{2} z\) has been verified.

Step 1: Write the Identity

The given identity is \(3+\sin ^{2} z=4-\cos ^{2} z\). The goal is to manipulate one side of this equation to look like the other side.

Step 2: Use the Pythagorean Identity

The Pythagorean identity is \(\sin^{2}z + \cos^{2}z = 1\). If we solve for \(\cos ^{2} z\) we get \(\cos ^{2} z = 1 - \sin^{2}z\).

Step 3: Substitute into the Original Equation

Substitute \(\cos ^{2} z\) in the original equation with \((1 - \sin^{2}z)\), this simplifies to \(3+\sin ^{2} z=4-(1 - \sin^{2}z)\). Expanding the right side of the equation, we get \(3+\sin^{2} z= 4 - 1 + \sin^{2} z\), which simplifies to \(3+\sin^{2} z= 3 + \sin^{2} z\).

Step 4: Check if Both Sides Are Equal

In this step the equality is checked and we can see that both sides of the equation are indeed equal. Thus the identity has been verified.

Key Concepts

Pythagorean identity

In trigonometry, the Pythagorean identity is a fundamental concept due to its simplicity and utility. Whenever you see trigonometric equations involving \(\sin^2 z\) or \(\cos^2 z\), this identity can be utilized to make the solution easier. The identity states that \(\sin^2 z + \cos^2 z = 1\).
This equation helps in transforming and simplifying trigonometric expressions.
Here are some useful manipulations derived from the Pythagorean identity:

• If you need an expression for \(\sin^2 z\), you can rearrange it as \(\sin^2 z = 1 - \cos^2 z\).
• Conversely, to find \(\cos^2 z\), use \(\cos^2 z = 1 - \sin^2 z\).

These transformations are especially helpful for verifying identities, like our original exercise, where we were asked to verify the equation \(3 + \sin^2 z = 4 - \cos^2 z\). By understanding the Pythagorean identity, we can simplify and verify the equation more efficiently.

sine and cosine functions

The sine and cosine functions are central in trigonometry, representing the relationship between angles and ratios of the sides of a right triangle. Here’s what you might like to know about them:

• Sine Function (\(\sin z\)): This function calculates the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• Cosine Function (\(\cos z\)): Similarly, this function gives the ratio of the length of the adjacent side to the hypotenuse.

On the unit circle, these functions are crucial for defining angles and can easily express many trigonometric identities.
By using the Pythagorean identity (\(\sin^2 z + \cos^2 z = 1\)), we can relate these functions and manipulate their expressions easily.
As demonstrated in our exercise, the sine and cosine relationships are substituted without losing accuracy, simplifying the given identity.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying expressions to achieve a desired equation or form. It encompasses a variety of operations such as:

• Substituting equivalent expressions.
• Expanding or factoring out terms.
• Combining like terms.

In the context of our problem, we used algebraic manipulation to equate the two sides of the identity \(3 + \sin^2 z = 4 - \cos^2 z\). By substituting \(\cos^2 z\) with \(1 - \sin^2 z\), we can transform the equation to see that it naturally simplifies down to an equality (i.e., both sides are rearranged to be exactly the same).
This technique is commonly used in verifying trigonometric identities and requires a good grasp of both identities and algebraic intervention.
Whether simplifying complex expressions or verifying identities like we did, the role of algebraic manipulation is indispensable in trigonometry.