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Define the Pythagorean Identity.
The fundamental trigonometric identity: $\sin^2(x) + \cos^2(x) = 1$.
Define tangent in terms of sine and cosine.
$\tan(x) = \frac{\sin(x)}{\cos(x)}$
Define secant in terms of cosine.
$\sec(x) = \frac{1}{\cos(x)}$
Define cotangent in terms of sine and cosine.
$\cot(x) = \frac{\cos(x)}{\sin(x)}$
Define cosecant in terms of sine.
$\csc(x) = \frac{1}{\sin(x)}$
Explain the relationship between the Pythagorean theorem and the identity $\sin^2(x) + \cos^2(x) = 1$.
The identity is derived from the Pythagorean theorem applied to the unit circle, where $\sin(x)$ and $\cos(x)$ represent the y and x coordinates, respectively, and the radius is 1.
Explain how to derive the identity $1 + \tan^2(x) = \sec^2(x)$ from the basic Pythagorean identity.
Divide the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ by $\cos^2(x)$ to obtain $\tan^2(x) + 1 = \sec^2(x)$.
Explain how to derive the identity $1 + \cot^2(x) = \csc^2(x)$ from the basic Pythagorean identity.
Divide the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ by $\sin^2(x)$ to obtain $1 + \cot^2(x) = \csc^2(x)$.
What is the significance of sum and difference identities?
They allow us to find trigonometric values of angles that are sums or differences of known angles.
What is the significance of double-angle identities?
They allow us to find trigonometric values of angles that are twice the size of a known angle.
What is the Pythagorean identity relating tangent and secant?
$1 + \tan^2(x) = \sec^2(x)$
What is the Pythagorean identity relating cotangent and cosecant?
$1 + \cot^2(x) = \csc^2(x)$
What is the sine sum identity?
$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
What is the cosine sum identity?
$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
What is the sine difference identity?
$\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$
What is the cosine difference identity?
$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$
What is the sine double-angle identity?
$\sin(2a) = 2\sin(a)\cos(a)$
What is the cosine double-angle identity (first form)?
$\cos(2a) = \cos^2(a) - \sin^2(a)$
What is the cosine double-angle identity (second form)?
$\cos(2a) = 1 - 2\sin^2(a)$
What is the cosine double-angle identity (third form)?
$\cos(2a) = 2\cos^2(a) - 1$