Unit 4 Advanced Algebra Review

Unit Circle

• Centered at the origin with a radius of one, providing key coordinates for trigonometric functions.

Unit Circle Coordinates

• Coordinates represented as (x,y) correspond to (cos, sin) for angle measurement.

Angles

• Formed by two rays sharing a common endpoint.

Coterminal Angles

• Share the same terminal side in standard position; can be found by adding or subtracting 360° for degrees or 2π for radians.

Degrees and Radians

• Degrees represented as decimals, while radians as fractions.

Conversion from Degrees to Radians

• Multiply degrees by π/180 to convert to radians.

Conversion from Radians to Degrees

• Multiply radians by 180/π to convert to degrees.

Sine and Cosecant Relations

• sin(θ) = y/r and csc(θ) = r/y, establishing the relationship between sine and cosecant.

Cosine and Secant Relations

• cos(θ) = x/r and sec(θ) = r/x, linking cosine with secant.

Tangent and Cotangent Relations

• tan(θ) = y/x and cot(θ) = x/y, demonstrating the connection between tangent and cotangent.

Trigonometry Basics

• Remember the mnemonic SOH CAH TOA for defining sine, cosine, and tangent.

Reference Angle

• The acute angle formed between the terminal side of an angle and the horizontal axis, crucial for evaluating trigonometric functions.

Graphing Trigonometric Equations

• For y=sin(x), observe the y-values; for y=cos(x), focus on the x-values; for y=tan(x), calculate by dividing y-value by x-value.

Changing Trigonometric Function Periods

• Adjustments can be made through the equation y=cos(1/2 θ), where the original period is doubled.

Phase Shift

• Refers to the horizontal movement of a trigonometric function, expressed in equations like y=cos(x - c).

Order of Operations for Equations

• Follow proper sequence in equations such as y= -2sin(1/2 (x - d)).

Trigonometric Identities

• Key identities include:
  • sin²(θ) + cos²(θ) = 1
  • tan(θ) = sin(θ)/cos(θ)
  • Derivations leading to single value outcomes.

Key Sine and Cosine Values

• sin(90°)=1; sin(-90°)=-1; sin(30°)=1/2; cos(60°)=1/2.
• Notable angles yield significant sine and cosine values.

Co-Function Identities

• Not explicitly defined in the notes but pertain to the relationships between angles.

Negative Angle Identities

• Identifying behavior of trigonometric functions under negative angles:
  • sin(-x) = -sin(x)
  • cos(-x) = cos(x)
  • tan(-x) = -tan(x)

Verifying Trigonometric Equations

• For verification, manipulate one side of an equation using trigonometric identities to match the other side.

Solving Trigonometric Equations

• Tackle like linear equations, isolating trig functions (sin(x), cos(x), tan(x)) on one side of the equation.

Tangent Values

• Understand tangent values in reference to the unit circle:
  • (1/2, 3/2) yields 3; (2/2, 2/2) yields 1; (3/2, 1/2) yields 3/3.
• Positive values occur in quadrants 1 and 3, while negative values are found in quadrants 2 and 4.