Problem 25 - Precalculus: Functions and Graphs

Question

Verify the identity. \(2 \sin ^{2} 2 t+\cos 4 t=1\)

Step-by-Step Solution

Answer

Yes, the identity is verified: both sides simplify to 1.

Step 1: Recognize Trigonometric Identities

First, we need to recognize the identity that will help us simplify the expression. Note that one useful identity is the double angle identity: \(\cos 2\theta = 1 - 2\sin^2 \theta\). In this problem, the angle for \(\cos\) is \(4t\) which can be expressed as \(2 \times 2t\).

Step 2: Apply the Cosine Double Angle Identity

Use the identity \( \cos 2\theta = 1 - 2\sin^2 \theta \) for \(\cos 4t\). We have \(\cos 4t = \cos(2 \times 2t)\), therefore:\[\cos(4t) = 1 - 2\sin^2(2t)\]

Step 3: Substitute and Simplify

Substitute \(\cos 4t\) from step 2 into the original equation:\[2\sin^2(2t) + (1 - 2\sin^2(2t)) = 1\]When you simplify this expression, the \(2\sin^2(2t)\) from both the positive and negative terms cancel each other out.

Step 4: Confirm the Simplification

After canceling out the terms:\[1 = 1\]This shows that both sides of the identity are equal.

Key Concepts

Double Angle Formulas

In trigonometry, double angle formulas are powerful tools used to simplify the expressions involving trigonometric functions of double angles.
Double angle formulas help in expressing trigonometric functions like sine, cosine, and tangent as functions of half the angle.
One of the most common double angle identities used is for the cosine function: \( \cos 2\theta = 1 - 2\sin^2 \theta\) or \(\cos 2\theta = 2\cos^2 \theta - 1\).
These identities are particularly useful when verifying identities or solving trigonometric equations by breaking down complex expressions.

For instance, in the exercise above, the angle was \(4t\), and we used the identity for cosine of double angles: \(\cos(4t) = \cos(2 \times 2t)\). This helped us express \(\cos 4t\) as \(1 - 2\sin^2(2t)\), which simplified the given identity.
By knowing and applying these formulas, you can effortlessly convert complex trigonometric expressions into simpler forms.

Trigonometric Equations

Trigonometric equations, like algebraic equations, involve trigonometric functions and constants.
Solving these equations often require using identities, rearranging terms, and finding the values of angles that satisfy the equation.

Consider the original exercise, which was presented as a trigonometric identity: \(2 \sin^2 2t + \cos 4t = 1\).
At first glance, it may seem daunting. However, by utilizing trigonometric identities like the double angle formulas, we can transform and simplify the equation.

Here is how it works: the task was to demonstrate that both sides of the equation are indeed equivalent, verifying it as a true identity.
By substituting the double angle formula of \(\cos 4t\) into the equation, it was simplified to \(2\sin^2(2t) + (1 - 2\sin^2(2t))\), which clearly equals 1 showing its validity.
Understanding these types of equations is crucial for students in trigonometry, as they are foundational in more advanced topics.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and plays a key role in trigonometry.
It represents the x-coordinate of a point on the unit circle as you move around it.
Important properties of the cosine function include its periodicity, being even-symmetric about the y-axis, and having values ranging from -1 to 1.

For instance, the cosine of an angle is found by computing the adjacent side over the hypotenuse in a right-angled triangle.

• The cosine of zero degrees is 1.
• The cosine of 90 degrees is 0.

In our exercise, the cosine function appeared as \(\cos 4t\).
We utilized the double angle identity to express this function in a simpler form, \(1 - 2\sin^2(2t)\), making it easier to solve the trigonometric equation.
By understanding and manipulating the properties of the cosine function, particularly through identities, students can gain deeper insights into solving complex trigonometric problems.