How does the period of $y = \cos(2x)$ compare to the period of $y = \cos(x)$?

The period of $y = \cos(2x)$ is half that of $y = \cos(x)$. Doubling the angle frequency compresses the graph horizontally, meaning it completes one full cycle in $\pi$ (or $180^\circ$) instead of $2\pi$.

When solving an equation containing both $\cos(2\theta)$ and $\cos(\theta)$, which substitution is most effective?

Substitute $\cos(2\theta)$ with $2\cos^2(\theta) - 1$. This creates a quadratic equation entirely in terms of $\cos(\theta)$, which can then be solved using standard factoring or the quadratic formula.

What is the fundamental difference between the identity $\cos^2 A + \sin^2 A$ and the formula for $\cos(2A)$?

$\cos^2 A + \sin^2 A$ is the Pythagorean identity which always equals $1$, regardless of the angle. In contrast, $\cos(2A) = \cos^2 A - \sin^2 A$ is a double angle formula that changes value depending on the angle $A$.

What is a common mistake when expanding $\sin(2A)$?

A frequent error is assuming $\sin(2A) = 2\sin A$. The correct expansion is $2\sin A \cos A$, which accounts for the interaction between the sine and cosine components of the angle.

Why is it dangerous to use $\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}$ without checking the value of $A$?

If $A = 45^\circ$ or $135^\circ$, then $\tan^2 A = 1$, making the denominator zero. This results in an undefined value, corresponding to the vertical asymptotes of the tangent function at $90^\circ$ and $270^\circ$.

What error occurs if you forget to adjust the search interval when solving $\sin(2x) = 0.5$ for $0 \le x < 180^\circ$?

You might miss valid solutions. Because the argument is $2x$, you must look for all values of $2x$ in the range $0 \le 2x < 360^\circ$ before dividing by $2$ to find the final values for $x$.

State the three equivalent forms for $\cos(2A)$.

1) $\cos^2 A - \sin^2 A$ 2) $2\cos^2 A - 1$ 3) $1 - 2\sin^2 A$. These are derived using the Pythagorean identity.

What is the double angle formula for $\tan(2A)$?

$\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}$. This formula is derived by setting $B=A$ in the tangent addition formula $\frac{\tan A + \tan B}{1 - \tan A \tan B}$.

How can you express $\sin A \cos A$ as a single trigonometric function?

$\sin A \cos A = \frac{1}{2}\sin(2A)$. This is obtained by rearranging the standard double angle formula $\sin(2A) = 2\sin A \cos A$.

How is the $\sin(2A)$ formula derived?

It is derived from the addition formula $\sin(A+B) = \sin A \cos B + \cos A \sin B$ by letting $B = A$. This results in $\sin(A+A) = \sin A \cos A + \cos A \sin A = 2\sin A \cos A$.

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Trigonometry

Double Angle Formulae

Summary

Double angle formulae are trigonometric identities that express functions of an angle $2A$ in terms of functions of the single angle $A$ . These identities are fundamental tools in trigonometry used to simplify expressions, solve equations involving multiple frequencies of an angle, and linearize squared trigonometric terms for calculus applications.

1. Definition & Core Concepts

Double angle formulae are a specific subset of trigonometric identities where the argument of the function is exactly twice that of the constituent parts.
The primary identities involve the three main trigonometric ratios: sine, cosine, and tangent.
These formulae allow for the 'unfolding' of a double angle into single-angle components, which is essential when an equation contains a mixture of terms like $\sin(2\theta)$ and $\sin(\theta)$ .

Graph comparing sin(theta) and sin(2theta), illustrating how the double angle doubles the frequency of the wave.

2. Underlying Principles

3. The Three Forms of Cosine

4. Methods & Techniques

5. Key Distinctions

6. Exam Strategy & Tips