Unit Circle Equation
$$x^2 + y^2 = 1$$
All points \((x, y)\) on the unit circle satisfy this equation. The radius is always 1.
Degrees & Radians
- $$360^\circ = 2\pi \text{ radians}$$
- $$1^\circ = \frac{\pi}{180} \text{ radians}$$
- $$1 \text{ radian} = \frac{180^\circ}{\pi}$$
Key Angles on the Unit Circle
| Angle | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| \(0°\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(30°\) | \(π/6\) | \(1/2\) | \(\sqrt{3}/2\) | \(1/\sqrt{3}\) |
| \(45°\) | \(π/4\) | \(\sqrt{2}/2\) | \(\sqrt{2}/2\) | \(1\) |
| \(60°\) | \(π/3\) | \(\sqrt{3}/2\) | \(1/2\) | \(\sqrt{3}\) |
| \(90°\) | \(π/2\) | \(1\) | \(0\) | undefined |
Pythagorean Identity
$$\sin^2\theta + \cos^2\theta = 1$$
Basic Trig Functions
- $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
- $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
- $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$$
Reference Angles & Signs
- Reference angle: The acute angle formed with the \(x\)-axis.
- Signs: All Students Take Calculus (ASTC) mnemonic:
- All trig functions positive in Quadrant \(I\)
- Sine & Cosecant positive in Quadrant \(II\)
- Tangent & Cotangent positive in Quadrant \(III\)
- Cosine & Secant positive in Quadrant \(IV\)
Arc Length Formulas
- Angle in degrees: $$\ell = \frac{\theta}{360^\circ} \cdot 2\pi r$$
- Angle in radians: $$\ell = r\theta$$
Coterminal Angles
- Degrees: $$\theta + 360^\circ \times k$$
- Radians: $$\theta + 2\pi k$$
- where \(k\) is any integer
Even-Odd & Symmetry Identities
- $$\sin(-\theta) = -\sin\theta$$
- $$\cos(-\theta) = \cos\theta$$
- $$\tan(-\theta) = -\tan\theta$$