Unit 4 Advanced Algebra Review

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Questions and Answers

What is the definition of a Unit Circle?

A circle with variable radius

A circle centered at any point on the plane

A circle whose center is at the origin and has a radius of one (correct)

A circle with a radius of two

How do you read Unit Circle coordinates?

(x,y)=(cos,sin)

An angle is formed by two rays with the same endpoint and makes an _______.

angle

What are coterminal angles?

Two angles in standard position with the same terminal side. Signup and view all the answers

Match Degrees with their equivalent in Radians.

Degrees = = Decimals Radians = = Fractions Signup and view all the answers

How do you convert from Degrees to Radians?

Multiply by π/180 Signup and view all the answers

How do you convert from Radians to Degrees?

Multiply by 180/π Signup and view all the answers

What is the formula for sin(0)?

y/r Signup and view all the answers

What is the formula for csc(0)?

r/y Signup and view all the answers

What is the formula for cos(0)?

x/r Signup and view all the answers

What is the formula for sec(0)?

r/x Signup and view all the answers

What is the formula for tan(0)?

y/x Signup and view all the answers

What is the formula for cot(0)?

x/y Signup and view all the answers

What does SOH CAH TOA stand for in Trigonometry?

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent Signup and view all the answers

What is a Reference Angle?

The positive acute angle formed by the terminal side and the horizontal axis. Signup and view all the answers

What does the equation y = sin(x) represent in graphing trig equations?

Look at y-values. Signup and view all the answers

What is the goal for changing the period of a trigonometric function?

Adjusting the function to manipulate its cycle length. Signup and view all the answers

What is Phase Shift in trigonometry?

When a trigonometric function moves to the left or right. Signup and view all the answers

What are Trig Identities?

Equations that must hold true for all angles under consideration. Signup and view all the answers

What is the method for verifying trig equations?

Manipulating one side to appear like the other side using identities. Signup and view all the answers

How do you solve trig equations?

Get the unknown trig function on one side and everything else on the other side. Signup and view all the answers

The tangent value for (1/2, 3/2) is _______.

3 Signup and view all the answers

The tangent value for (2/2, 2/2) is _______.

1 Signup and view all the answers

The tangent value for (3/2, 1/2) is _______.

3/3 Signup and view all the answers

In which quadrants are tangent values positive?

Quadrants 1 and 3. Signup and view all the answers

In which quadrants are tangent values negative?

Quadrants 2 and 4. Signup and view all the answers

Study Notes

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.

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Test your knowledge with this comprehensive flashcard set on Unit 4 of Advanced Algebra. Covering key concepts like the unit circle, angle definitions, and coterminal angles, these flashcards are perfect for review and reinforcement. Challenge yourself and solidify your understanding of these important mathematical topics.