Properties And Principles Of Trigonometry

Unit  – TRIANGLES & TRIGONOMETRY

Chapter: – Properties & Principal of Trigonometry

What students will learn in this Section

In the Trigonometry & Triangles section of the SAT, students delve into the intricacies of triangle geometry. They discern the distinguishing features of Equilateral, Isosceles, and Scalene triangles, grasping not only their side-length characteristics but also the corresponding angle properties. The introduction of trigonometric functions—Sine, Cosine, and Tangent—equips students with tools to navigate the relationships between sides and angles in right-angled triangles.

Beyond triangles, students explore the sum of interior angles in triangles and the properties of angles within quadrilaterals. This comprehensive understanding enables them to approach a myriad of geometry problems presented in the SAT, fostering critical thinking and analytical skills essential for success in the Math section.

Important Definitions:

• Equilateral Triangle:
  • A triangle with all three sides of equal length.
• Isosceles Triangle:
  • A triangle with at least two sides of equal length.
• Scalene Triangle:
  • A triangle with all three sides of different lengths.
• Sine (sin):
  • In a right-angled triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.
• Cosine (cos):
  • In a right-angled triangle, the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
• Tangent (tan):
  • In a right-angled triangle, the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
• Parallelogram:
• A quadrilateral with opposite sides equal and parallel.
• Rectangle:
• A parallelogram with all angles equal to 90 degrees.
• Rhombus:
• A parallelogram with all sides equal.

Important Formulae:

1. Lines and Angles:
   • Slope of a Line (m): m=
   • Distance Formula between two points (P₁(x₁, y₁) and P₂(x₂, y₂)):

1. Complementary and Supplementary Angles:
   • Complementary Angles: ∠A+∠B=90∘
   • Supplementary Angles:  ∠C+∠D=180∘
2. Triangles:
   • Sum of Interior Angles of a Triangle: Sum=180∘
   • Pythagorean Theorem (for a right-angled triangle ABC with hypotenuse c): a²+b²=c²
3. Special Right Triangles:
   • 45-45-90 Triangle: If the acute angles are both 45 degrees, then the sides are in the ratio 1:1:√2.
   • 30-60-90 Triangle: If the angles are 30, 60, and 90 degrees, then the sides are in the ratio 1:√3:2.
4. Area Formulas:
   • Area of a Triangle (given base b and height h): Area= ​×b×h
   • Area of a Right-Angled Triangle (given legs a and b): Area=​ ×a×b

Speed Strategy

1. Memorize Key Formulas:
   • Memorize essential formulas to reduce the time spent looking them up. This includes formulas for Area, sector Angles, Tangent line equation, and other geometrical measures.
2. Practice Formula Rearrangement:
   • Familiarize yourself with rearranging formulas. This skill allows you to quickly solve for different variables without having to derive the entire formula.
3. Use Pre-calculated Constants:
   • Pre-calculate constants or values that frequently appear in formulas. For example, memorize common Z-scores or values associated with circles & Angles.