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.
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.
$\csc(x) = \frac{1}{\sin(x)}$
How do you simplify an expression like $2\sin^2(x) + 2\cos^2(x) - 1$?
1. Factor out the 2: $2(\sin^2(x) + \cos^2(x)) - 1$. 2. Apply the Pythagorean identity: $2(1) - 1$. 3. Simplify: $2 - 1 = 1$.
How do you find $\sin(75)$ using sum identities?
Express 75 as 45 + 30. Use the sine sum identity: $\sin(45 + 30) = \sin(45)\cos(30) + \cos(45)\sin(30)$. Evaluate.
How to solve for $\cos(x)$ given $\sin(x) = \frac{3}{5}$ and $x$ is in the first quadrant?
1. Use the Pythagorean identity: $\cos^2(x) = 1 - \sin^2(x)$. 2. Substitute: $\cos^2(x) = 1 - (\frac{3}{5})^2 = \frac{16}{25}$. 3. Solve for $\cos(x)$: $\cos(x) = \frac{4}{5}$ (positive since x is in the first quadrant).
How to find $\sin(2x)$ if $\sin(x) = 0.6$ and $\cos(x) = 0.8$?
1. Use the double-angle identity: $\sin(2x) = 2\sin(x)\cos(x)$. 2. Substitute: $\sin(2x) = 2(0.6)(0.8)$. 3. Calculate: $\sin(2x) = 0.96$.
How to simplify $\cos(a + b) - \cos(a - b)$?
1. Expand using sum and difference identities: $[cos(a)\cos(b) - \sin(a)\sin(b)] - [cos(a)\cos(b) + \sin(a)\sin(b)]$. 2. Simplify: $-2\sin(a)\sin(b)$.
