Complex Numbers

This math unit begins with foundational trigonometry, starting with identifying different sides of a right triangle—opposite, adjacent, and hypotenuse. It progresses to understanding and labeling the side ratios in right triangles, using the SOHCAHTOA acronym to deepen the comprehension of trigonometric relationships such as sine, cosine, and tangent. As students become more comfortable with these definitions, they move on to applying and manipulating these ratios to solve for unknowns in trigonometric expressions and focusing on approximating trigonometric values from diagrams. The unit further advances into calculating and approximating side length ratios and explicit side lengths using trigonometric principles, scaling up to more complex applications where geometric figures and ratios are used extensively. Towards the end, the unit incorporates fact families within trigonometry to link geometric and algebraic representations, fostering a comprehensive grasp of the relationships between angles, sides, and trigonometric identities.Skills you will learn include:

• Ratios of line lengths
• Opposite, adjacent, hypotenuse
• Sin, cos, tan
• Solving triangles with trignometry

This math unit initially focuses on understanding and calculating ratios of line lengths in various geometric configurations, including parallel and right angle line displays. It introduces students to basic trigonometry through the practical application of calculating these ratios and progresses to more complex tasks. As the unit develops, students begin working with trigonometric identities and ratios such as sine, cosine, and tangent. They learn to solve for unknown angles based on given side lengths and to express relationships using trigonometric formulas. The later sections of the unit continue to emphasize interpreting and solving problems using trigonometric ratios and identities but introduce more complex applications, involving decimal representations and extracting trigonometric values from diagrams. The unit concludes with a comprehensive understanding of using trigonometric principles to solve geometric problems, setting a strong foundation in trigonometry by systematically building fluency from simple ratio calculations to complex trigonometric applications.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry

In this math unit, students begin by learning the fundamental skills of converting angles between degrees and radians using the unit circle. They start with basic conversions, practicing both from degrees to radians and vice versa. As they progress, students further explore these conversions with a wide range of specific angles, enhancing their fluency and comprehension of angular measurements in both units. Midway through the unit, the focus shifts towards applying this knowledge to calculate the trigonometric values sine and cosine for angles defined in both radians and degrees. Students practice identifying the coordinates of points on the unit circle based on these angles. This includes exercises that provide angles in radians and degrees, for which they have to determine the cosine and sine coordinates. Towards the unit's conclusion, learners refine their understanding of mapping these angles and trigonometric coordinates visually and numerically on the unit circle. This involves more complex problem sets where they are required to interpret trigonometric functions and their corresponding coordinates or visualize angles based on provided coordinates, solidifying their comprehension of trigonometry’s relationship with the unit circle.Skills you will learn include:

• Unit circle
• Radians and degrees
• Cos and Sin values

This math unit begins by introducing students to the foundational concepts of trigonometry, focusing initially on understanding and using the basic trigonometric ratios—sine, cosine, and tangent—associated with right triangles. Early topics cover identifying trigonometric relationships and learning how to formulate correct trigonometric expressions based on given side lengths or angles. Progressively, the unit moves into more practical applications, teaching students to approximate these trigonometric ratios from diagrams and visual representations, further ingraining the fundamentals. The problems evolve to include calculating unknown angles from given side ratios, both in fraction and decimal forms, utilizing trig identities to facilitate these computations. Towards the latter part of the unit, the complexity increases as students apply their acquired skills to solve for unknown side lengths, angles from diagrams, and exact trigonometric values. Each successive topic builds on prior knowledge, culminating in more advanced practice that combines theoretical trigonometric principles with practical problem-solving skills, reinforcing understanding and application of trigonometry in geometric contexts.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry

This math unit centers around advanced trigonometry skills, emphasizing the use of the unit circle to calculate and understand the trigonometric functions sine and cosine for angles measured both in degrees and radians. Initially, the unit introduces learners to identifying sine and cosine as coordinates corresponding to specified angles on the unit circle, advancing to recognize and calculate these values directly from visual unit circle representations. As the unit progresses, students delve deeper into solving for trigonometric coordinates using both specific degrees and radians, enhancing their ability to visualize and translate between angle measures and coordinate points on the circle. Further topics solidify understanding by connecting coordinates back to angle measurements in radians, transitioning into the application of Pythagorean identities within the unit circle context. The final sections focus on deepening learners’ grasp of trigonometric identities, particularly how to convert between the dimensions or coordinates of points on the unit circle and their corresponding sine or cosine values through the Pythagorean theorem, underlining fundamental relationships and properties within trigonometry. This progression from basic identification to complex applications underscores a comprehensive mastery of the trigonometric unit circle.Skills you will learn include:

• Unit circle
• Trig identities
• Pythagorean identities
• Negative angle identities

This math unit begins by establishing foundational skills in identifying trigonometric identities and calculating angles and ratios from descriptions and diagrams. Initially, students focus on understanding basic trig identities, such as sine, cosine, and tangent, progressing into the practical application of these ratios to calculate angles using inverse functions and descriptions in word and arc notation. The unit advances to converting given ratios and angles into trigonometric values, both in decimal and exact forms, emphasized through the use of diagrams which help in visual understanding and approximation. Later, the unit advances to solving for unknown side lengths and angles in various triangles, demonstrating applications in geometry through practical problems. This progression solidifies the student's ability to manipulate and apply trigonometric principles in diverse scenarios, reinforcing comprehensive understanding of trigonometry from theoretical identities to their practical applications in solving geometric problems.Skills you will learn include:

• Working with ratios
• Sin, cos, tan
• Solving triangles with trignometry

In this math unit on trigonometry fundamentals, students begin with basic skills such as solving trigonometric ratios (sine, cosine, tangent) using given side values of right triangles, and then advance to calculating unknown angles from side measurements. As they progress, they apply these ratios to solve for missing side lengths in right triangles using angles and side measurements provided, enhancing their problem-solving skills in practical scenarios. Further into the unit, students tackle more complex tasks such as solving for angles and sides from detailed diagrams, as well as computing and approximating trigonometric ratios and angles from visual information and decimal representations. This steadily builds their ability to interpret and analyze geometric data and apply trigonometric identities in more advanced problems. Towards the end of the unit, the focus shifts to refining the understanding of trigonometric relationships and identities through problems involving ratios described in both fraction and decimal forms, extracted or approximated from diagrams. This comprehensive approach ensures a deep and practical mastery of trigonometry principles, equipping students for further mathematical challenges.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry
• Solving angles
• Solving side lengths

This math unit begins with foundational trigonometry skills, focusing on calculating trigonometric ratios given triangle side lengths and progressing to determine angles using these ratios. Students practice solving for unknown side lengths and angles from basic numerical data and then advance to solving from diagrams. As the unit progresses, the complexity increases as students engage with specific trigonometric rules—the Rule of Sines and Rule of Cosines—starting with setups and moving into full calculations. These principles are applied to solve for side lengths and angles in various triangles, including non-right triangles. Towards the end of the unit, students use trigonometric functions to solve for practical aspects like the areas of triangles, utilizing these more advanced concepts to setup and compute areas using given sides and angles. This progression builds a comprehensive skill set in solving common and complex trigonometric problems in geometry.Skills you will learn include:

• Law of Sines
• Law of Cosines
• Solving triangles and triangle area with trignometry

This math unit begins with students learning to apply Heron's Formula in the context of solving triangles. Initially, the focus is on selecting the correct formula for computing the area of triangles based on given side lengths and preparing them for application by understanding different scenarios and their requirements. As the unit progresses, learners move towards performing full calculations using Heron's Formula. They solve problems to determine the area of various triangles using precise triangulation specifics and rigorous application of the trigonometric relationships and geometry principles. This sequence from setup to full application hones the students' ability to manipulate and use trigonometric formulas effectively, improving their problem-solving skills within the realm of trigonometry.Skills you will learn include:

• Heron's formula
• Sin, cos, tan
• Solving triangles with trignometry

This math unit progresses from basic to advanced concepts involving complex numbers. Initially, students focus on rewriting and simplifying complex numbers, starting with square roots containing negative radicands. They practice expressing these roots in terms of complex numbers and simplifying radicals. As the unit advances, students engage with operations on the complex plane, such as subtraction and division of complex numbers, specifically focusing on imaginary parts. To deepen understanding, they also simplify and evaluate powers of the imaginary unit, \(i\). Later, the unit moves toward geometric interpretations and transformations of complex numbers, including calculating absolute values and converting between polar and rectangular forms in both radians and degrees. They apply trigonometric functions and Euler’s formula, essential for converting complex numbers' exponential form into rectangular form. These cumulative skills build a comprehensive understanding of complex number manipulation, encompassing algebraic operations and geometric interpretations.Skills you will learn include:

• Meaning of i
• Complex plane
• Polar coordinates