Sin, Cos, Tan, Cot, Sec, Csc:
Definitions of common trigonometric functions related to angles.
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)
cot(θ) = 1 / tan(θ)
sec(θ) = 1 / cos(θ)
csc(θ) = 1 / sin(θ)
Standard angles and their trig functions:
θ = 0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
θ = 30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = √3/3
θ = 45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1
θ = 60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3
θ = 90°: sin(90) = 1, cos(90) = 0, tan(90) = Undefined
Evaluate expressions like:
sin(25°), cos(60°), etc., using known values or calculators.
Use identities for simplification:
cot(θ) = 1/tan(θ), csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ)
Example: cot(25°) = 1/tan(25°)
Complementary Angles: sin(θ) = cos(90° - θ)
Input Adjustments: cot(40°) can be rearranged as csc(50°)
**Use of half-angle and double-angle identities.
Formulas like: 4 cos(45°) - 2 sin(45°) can be resolved to find respective values.
Solve equations step-by-step, combining like terms and using known angles.
Law of Sines: (a/sin(A)) = (b/sin(B)) = (c/sin(C))
Law of Cosines: c² = a² + b² - 2ab*cos(C)
Use to determine lengths and angles in triangles.
Example: Given angles and sides, calculate unknowns using these laws.
For each calculation, use trigonometric identities appropriately:
Simplify products and ratios of sine and cosine functions.
Convert to decimal if necessary and isolate the variable if solving for it.
When necessary, refer back to properties like:
Sum and difference formulas (e.g., sin(A ± B), cos(A ± B)).
Pythagorean identities to assist in calculations.