Plane Geometry and Trigonometry

Angles

Types:

• Acute: $< 90°$
• Right: $= 90°$
• Obtuse: $> 90°$ and $< 180°$
• Straight: $= 180°$

Complementary angles: Add to $90°$ Supplementary angles: Add to $180°$

Triangles

Angle Sum

All triangles: angles sum to $180°$

Pythagorean Theorem

For right triangles: $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse)

Special Right Triangles

45-45-90: $\text{sides: } x, x, x\sqrt{2}$

30-60-90: $\text{sides: } x, x\sqrt{3}, 2x$

Area

$A = \frac{1}{2}bh$

Circles

Circumference: $C = 2\pi r = \pi d$

Area: $A = \pi r^2$

Arc length: $\text{Arc} = \frac{\theta}{360°} \times 2\pi r$

Trigonometry

SOH-CAH-TOA

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Common Values

| Angle | sin | cos | tan | |-------|-----|-----|-----| | $30°$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ | | $45°$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ | | $60°$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |

Law of Sines (for any triangle)

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

ACT Tips

• Memorize special right triangles! They appear often
• Draw diagrams if not provided
• Use calculator for trig values (set to degrees!)
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17