Unit 7 Test Review Notes

Trigonometry Basics

1. Sine on the Unit Circle

• Point R on Unit Circle: Given a coordinate ((x, y)) on the unit circle, the sine of the angle is the (y)-coordinate.

2. Converting Angles

• Convert -560 degrees to Radians
  • Use the conversion formula: ( \text{radians} = \text{degrees} \times \frac{\pi}{180} ).

3. Sketching Angles

• Sketching (\theta) on Coordinate Plane
  • Identify the quadrant based on the angle.
  • Find the reference angle.

4. Quadrants and Angles

• Identify Quadrants
  • Use the given angle to determine in which quadrant the terminal side lies.
  • Calculate reference angles for angles such as 655 degrees.

5. Trigonometric Functions

• Identifying Trig Functions
  • Sine or Cosine based on the coordinates or given information.
• Sign of Functions
  • Quadrant where (\cos) is negative and (\sin) is positive.

6. Converting Radians to Degrees

• Conversion Formula: ( \text{degrees} = \text{radians} \times \frac{180}{\pi} ).

Graph Analysis

9. Graph Characteristics

• Graph Types: Sine or Cosine.
• Midline Equation: The horizontal line that the graph oscillates around.
• Amplitude: Half the distance between the maximum and minimum values.
• Period: The length of one complete cycle of the graph.
• B-Value: Affects the period of the function.
• Equation: General form ( y = a \sin(bx-c) + d ) or ( y = a \cos(bx-c) + d ).

10. Equation Analysis

• Understanding Given Equation: Identify amplitude, midline, and period from the equation.

11. Writing Equation from Graph

• Determine Midline, Amplitude, Period, Function Type (Sine/Cosine), and B-Value
• Construct the Equation

Applied Problems

12. Ferris Wheel Problem

• Period: 30 minutes for a full rotation.
• Maximum Height: 90 feet.
• Minimum Height: 2 feet.
• Time to Highest Point: Half the period of the Ferris wheel.

13. Beach Tide Cycle

• Identify Low Tide Times: Use the tide graph to find optimal shell-collecting times.

14. Waterwheel Trig Function

• Calculate Height at Specific Time: Use the function to find height at 18 seconds.
• Identify Extremes: Highest and lowest points on the waterwheel.
• Revolution Time: Time for complete cycle.

15. Graph Labeling

• Label Period, Amplitude, and Midline
• Use the graph to visually identify these components.