Precalculus Trigonometry and Graph Transformations

• The precalculus course at Nevada State College focuses on trigonometry, which is the study of triangles and their connections to circles.
• Bloom's Taxonomy is used to gauge the depth of learning, ranging from basic information recall to advanced idea generation.
• The course structure includes pre-class activities, in-class learning activities, and homework to build a comprehensive understanding of the topics.
• Math learning is emphasized as a process of connecting ideas and applying concepts rather than just memorizing rules.
• Angles are defined based on the intersection of line segments, with terms like vertex, initial side, and terminal side determining measurement direction.
• Angles can be measured in degrees or radians, with radians being preferred in mathematics due to their geometric significance.
• The unit circle, a circle with radius 1 centered at the origin, is crucial for understanding trigonometric functions.
• Special angles on the unit circle, like 0, π/2, π, 3π/2, and 2π, have specific coordinates used to calculate trigonometric functions.
• Trigonometric functions have periodicity (repetition at regular intervals) with a period of 2π due to the circular nature of the unit circle.
• Symmetry in trigonometric functions is observed through the relationship between positive and negative angles on the unit circle, impacting functions like sine, cosine, secant, and cosecant.- Trigonometry can be understood through right triangles where the two legs and the hypotenuse are defined, along with the relationships between them based on an angle.
• The Pythagorean theorem states that a² + b² = c² for right triangles, allowing to find the length of the third side if two sides are known.
• All six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) can be defined in terms of the relationships between sides of right triangles.
• Special right triangles (45-45-90 and 30-60-90) provide specific side lengths and values for trigonometric functions (sine, cosine, tangent) that are used in the unit circle.
• Reciprocal identities, quotient identities, and Pythagorean identities are important in trigonometry for deriving relationships between trigonometric functions.
• Understanding signs of trigonometric functions in different quadrants is crucial for determining their values based on given information.
• Reference angles in all four quadrants help determine trigonometric function values by relating them to angles in the first quadrant.
• Algebraic methods can be used to find trigonometric function values based on known information about a specific angle and quadrant.
• Graphs of sine and cosine functions show oscillating curves with key values at every PI/2, with the cosine graph being a shifted version of the sine graph.
• The periodic nature of sine and cosine functions with values constrained between plus/minus 1 and key values occurring regularly every PI/2.- Transformations can affect the shapes of curves, including parabolas, sine, and cosine functions, involving vertical and horizontal shifting, stretching, and flipping.
• The period of a function determines its repetition within regular intervals, while the fundamental period helps track shifting and stretching of the function's period.
• The midline of a function cuts it vertically in half and is primarily affected by vertical shifts.
• The amplitude of a function measures the vibrations' range from the midline to the maximum or minimum points.
• Graphing transformations involves identifying the fundamental period, determining key values, transforming corresponding y-coordinates, and plotting points.
• Shifting transformations involve adding or subtracting values to the function or its argument for horizontal or vertical shifts.
• Stretching transformations come from multiplying the entire function or argument by a value for horizontal or vertical stretches.
• The general transformation formula is y = a*f(bx - c) + d, with a specific order of operations: horizontal shift, horizontal stretch, vertical stretch, and vertical shift.
• Tangent functions have specific zeros and vertical asymptotes that help in graphing, following periodicity with a fundamental period of negative PI/2 to PI/2.
• Graphing reciprocals involves understanding that large X values result in small reciprocals and small X values lead to large reciprocals, except for +/-1.

Explore trigonometry concepts such as angles, trigonometric functions, special triangles, identities, signs in different quadrants, graphing sine and cosine functions, and transformations. Learn how to apply algebraic methods to find trigonometric function values and graph transformations to affect curve shapes.