The keyword cos 2 a refers to the cosine double-angle identity for \(\cos(2a)\). In algebra and trigonometry, this identity is essential for rewriting expressions involving \(\cos(2a)\) into powers of \(\sin a\) or \(\cos a\), which simplifies algebraic manipulation and equation solving.
Key identity \(\cos(2a)=\cos^2 a-\sin^2 a\).
Using \(\sin^2 a+\cos^2 a=1\), this produces two additional equivalent forms:
\[ \cos(2a)=2\cos^2 a-1 \quad\text{and}\quad \cos(2a)=1-2\sin^2 a. \]
Deriving the cos 2 a identity
Start from the cosine addition formula:
\[ \cos(x+y)=\cos x\cos y-\sin x\sin y. \]
Set \(x=a\) and \(y=a\). Then \(x+y=2a\), so:
\[ \cos(2a)=\cos(a+a)=\cos a\cos a-\sin a\sin a=\cos^2 a-\sin^2 a. \]
Generating equivalent algebraic forms
The Pythagorean identity \(\sin^2 a+\cos^2 a=1\) allows replacement of \(\sin^2 a\) or \(\cos^2 a\).
Form in terms of \(\cos a\) only
\[ \cos(2a)=\cos^2 a-\sin^2 a =\cos^2 a-(1-\cos^2 a) =2\cos^2 a-1. \]
Form in terms of \(\sin a\) only
\[ \cos(2a)=\cos^2 a-\sin^2 a =(1-\sin^2 a)-\sin^2 a =1-2\sin^2 a. \]
| Equivalent form for \(\cos(2a)\) | Most convenient when | Typical algebra goal |
|---|---|---|
| \(\cos(2a)=\cos^2 a-\sin^2 a\) | Both \(\sin a\) and \(\cos a\) appear | Factor or combine squared terms |
| \(\cos(2a)=2\cos^2 a-1\) | Expression is mostly in \(\cos a\) | Eliminate \(\sin^2 a\), solve for \(\cos a\) |
| \(\cos(2a)=1-2\sin^2 a\) | Expression is mostly in \(\sin a\) | Eliminate \(\cos^2 a\), solve for \(\sin a\) |
Algebraic use cases
1) Simplifying an expression
Simplify \(2\cos^2 a-1\). Recognize it as a double-angle expression:
\[ 2\cos^2 a-1=\cos(2a). \]
This replacement often reduces the number of terms and can expose patterns such as products-to-sums or periodic behavior.
2) Solving a trigonometric equation
Solve \(\cos(2a)=\tfrac{1}{2}\) for \(a\) on the interval \(0\le a<2\pi\). Use the angle-doubling substitution \(u=2a\), so \(0\le u<4\pi\):
\[ \cos u=\frac{1}{2}. \]
The solutions for \(\cos u=\tfrac{1}{2}\) are \(u=2\pi k\pm \frac{\pi}{3}\). Restricting to \(0\le u<4\pi\) gives:
\[ u\in\left\{\frac{\pi}{3},\ \frac{5\pi}{3},\ \frac{7\pi}{3},\ \frac{11\pi}{3}\right\}. \]
Divide by \(2\) to return to \(a\):
\[ a\in\left\{\frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{7\pi}{6},\ \frac{11\pi}{6}\right\}. \]
Visualization: geometric meaning of cos 2 a
The unit-circle diagram below shows angles \(a\) and \(2a\) from the positive x-axis. The x-coordinate of the point on the unit circle equals cosine, so the x-coordinate at angle \(2a\) visualizes \(\cos(2a)\). This helps connect the symbol cos 2 a with a concrete geometric interpretation.
Summary
The identity for cos 2 a is \(\cos(2a)=\cos^2 a-\sin^2 a\), with equivalent forms \(\cos(2a)=2\cos^2 a-1\) and \(\cos(2a)=1-2\sin^2 a\). These forms translate double-angle expressions into single-angle powers, a standard technique in algebra for simplifying trigonometric expressions and solving trigonometric equations.