Geometry
Master the most challenging geometry topics including advanced trigonometry and 3D geometry.
What You'll Learn
Advanced Trigonometry
Work with the unit circle, radian measure, and trigonometric identities.
3D Geometry
Calculate complex volumes, surface areas, and understand spatial relationships.
Complex Geometric Proofs
Apply geometric theorems and construct logical proofs.
Coordinate Geometry
Work with conic sections and complex coordinate plane problems.
Key Concepts
- •Radian Conversion: π radians = 180°, use for arc length and sector area
- •Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
- •Law of Cosines: c² = a² + b² - 2ab·cos(C)
- •Sphere Volume: V = (4/3)πr³ and Surface Area: SA = 4πr²
- •Pythagorean Identity: sin²θ + cos²θ = 1
Example Problem
A triangle has sides of length 7, 9, and 12. Find the measure of the largest angle.
Solution:
The largest angle is opposite the longest side (12).
Using Law of Cosines: c² = a² + b² - 2ab·cos(C)
12² = 7² + 9² - 2(7)(9)·cos(C)
144 = 49 + 81 - 126·cos(C)
144 = 130 - 126·cos(C)
14 = -126·cos(C)
cos(C) = -14/126 ≈ -0.111
C = arccos(-0.111) ≈ 96.4°
Study Tips
- ✓Use the Law of Sines for angle-side-angle or angle-angle-side scenarios
- ✓Use the Law of Cosines when you have side-side-side or side-angle-side
- ✓For 3D problems, break down complex shapes into familiar components
- ✓Remember that the unit circle relates angles to coordinates: (cos θ, sin θ)
Start Practicing
Ready to test your skills? Choose how you want to study: