Problem 23 - Algebra 2 and Trigonometry

Question

Show that \(\cos 2 \theta=1-2 \sin ^{2} \theta\)

Step-by-Step Solution

Answer

The identity \( \cos 2\theta = 1 - 2\sin^2 \theta \) is verified using the double angle and Pythagorean identities.

Step 1: Recall the Double Angle Identity

The double angle formula for cosine is: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta.\) This is one of the standard trigonometric identities.

Step 2: Express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \)

Recall the Pythagorean identity: \( \cos^2 \theta + \sin^2 \theta = 1. \) Rearrange this to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \): \( \cos^2 \theta = 1 - \sin^2 \theta. \)

Step 3: Substitute into the Double Angle Identity

Substitute \( \cos^2 \theta = 1 - \sin^2 \theta \) into the double angle identity for cosine: \(\cos 2\theta = (1 - \sin^2 \theta) - \sin^2 \theta.\)

Step 4: Simplify the Expression

Combine like terms in the equation: \(1 - \sin^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta.\) Thus, the identity \(\cos 2\theta = 1 - 2\sin^2 \theta \) is verified.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any value of the variable where both sides of the equation are defined. They form the foundation for solving trigonometric equations and simplifying expressions, which can appear daunting at first.
A few key trigonometric identities that are commonly used include:

• Sin and Cosine Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
• Sum-to-Product Identities: \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• Double Angle Formulas: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

Just like in our example problem, these identities help transform complex trigonometric expressions into a more manageable form. By becoming familiar with these identities, solving trigonometric equations becomes a more straightforward and intuitive task.
Remember, the logic behind these identities often parallels that of algebraic manipulations, so skills from algebra can be highly beneficial in understanding and applying trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, and it is derived from the Pythagorean theorem in geometry. The identity states:

• \(\cos^2 \theta + \sin^2 \theta = 1\)

This equation is crucial because it simplifies expressions and allows us to interchangeably use \(\sin^2 \theta\) and \(\cos^2 \theta\) in other forms.
To reframe \(\cos^2 \theta\) using the Pythagorean identity, rearrange it to find:

• \(\cos^2 \theta = 1 - \sin^2 \theta\)

This transformed version is especially useful when dealing with expressions involving double angle identities, as it allows substitutions that lead to simpler equations. For instance, in our initial exercise, replacing \(\cos^2 \theta\) facilitated the derivation of the identity \(\cos 2\theta = 1 - 2\sin^2 \theta\).
Developing an intuitive grasp of the Pythagorean identity can greatly ease the process of working through various trigonometric problems.

Cosine Function

The cosine function is a primary trigonometric function that represents the adjacent side's ratio over the hypotenuse in a right triangle for a given angle \(\theta\). It oscillates between -1 and 1 and is periodic with a period of \(2\pi\). This function is vital in understanding waves, oscillations, and rotations.
In the context of trigonometric identities, the cosine function can be expressed in several forms, which offers flexibility in solving problems. The double angle formula for cosine is a key identity:

• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

Using double angle identities, you can express trigonometric functions of multiple angles in terms of single angles, greatly simplifying the solution process, as seen in our example problem.
Additionally, understanding how to express \(\cos^2 \theta\) through other functions, such as rewriting it in terms of \(\sin^2 \theta\), enables you to solve problems that initially seem intractable. This capability makes the cosine function a powerful tool in the broader landscape of trigonometry.