IM3H Unit 8 Test Review KEY revised 3

Unit 8 Test Review

Trigonometric Functions

• 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)

Important Angle Values

• 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

Evaluating Trigonometric Expressions

• 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°)

Angle Relationships

• 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.

Solving Trigonometric Equations

• 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 and Cosines

• 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.

Problem-solving Steps

• 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.