Key Trigonometric Formulas
Basic Definitions
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent = sin(θ)/cos(θ)
Reciprocal Functions
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C) = 2R
Law of Cosines
c² = a² + b² - 2ab·cos(C)
Area Formulas
Area = (1/2)ab·sin(C)
Area = (1/2)bc·sin(A)
Area = (1/2)ac·sin(B)
Worked Examples
Example 1: Finding Missing Triangle Sides
Problem: In a right triangle, angle A = 30°, and the hypotenuse = 10. Find the opposite and adjacent sides.
Solution:
• opposite = hypotenuse × sin(A) = 10 × sin(30°) = 10 × 0.5 = 5
• adjacent = hypotenuse × cos(A) = 10 × cos(30°) = 10 × 0.866 = 8.66
Answer: opposite = 5, adjacent ≈ 8.66
Example 2: Using Law of Cosines
Problem: Triangle with sides a = 8, b = 6, and angle C = 60°. Find side c.
Solution:
• Using c² = a² + b² - 2ab·cos(C)
• c² = 8² + 6² - 2(8)(6)cos(60°)
• c² = 64 + 36 - 96(0.5)
• c² = 100 - 48 = 52
• c = √52 ≈ 7.21
Answer: c ≈ 7.21
Example 3: Angle of Elevation
Problem: A ladder 12 feet long leans against a wall. If the ladder makes a 65° angle with the ground, how high up the wall does it reach?
Solution:
• height = ladder length × sin(angle)
• height = 12 × sin(65°)
• height = 12 × 0.906
• height ≈ 10.87 feet
Answer: The ladder reaches approximately 10.87 feet up the wall.