1. Topic Overview
Main Concept/Theme: Trigonometric Functions
Key Learning Objectives:
– Understand and define the primary trigonometric functions (sine, cosine, and tangent).
– Learn the unit circle and how it relates to trigonometric functions.
– Apply the trigonometric identities for simplifying expressions.
– Solve trigonometric equations.
– Explore the graphs of trigonometric functions.
2. Key Terms and Definitions
- Sine (sin θ): A function that relates the angle in a right triangle to the ratio of the opposite side to the hypotenuse.
- Cosine (cos θ): A function that relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse.
- Tangent (tan θ): A function that relates the angle in a right triangle to the ratio of the opposite side to the adjacent side.
- Unit Circle: A circle with a radius of one unit centered at the origin of a coordinate plane, used for defining trigonometric functions.
- Radians: A way of measuring angles based on the radius of a circle.
- Amplitude: The maximum value of a trigonometric function.
- Period: The length of one complete cycle of a trigonometric function.
- Phase Shift: A horizontal shift left or right for the graph of a trigonometric function.
- Vertical Shift: A vertical displacement of the graph upwards or downwards.
3. Main Content Sections
Trigonometric Functions and Unit Circle
- Sine and Cosine Functions:
- Defined using the unit circle.
- ( sin(θ) ) is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
- ( cos(θ) ) is the x-coordinate of the same point.
- Tangent Function:
- Defined as the ratio ( tan(θ) = \frac{sin(θ)}{cos(θ)} ).
Graphs of Trigonometric Functions
- Graphing Sine and Cosine:
- ( y = sin(x) ): Starts at (0,0), reaches a maximum of 1 at ( \frac{π}{2} ), crosses zero at ( π ), reaches a minimum of -1 at ( \frac{3π}{2} ).
- ( y = cos(x) ): Starts at (0,1), crosses zero at ( \frac{π}{2} ), reaches a minimum of -1 at ( π ), crosses zero at ( \frac{3π}{2} ).
- Transformations:
- Amplitude changes: ( y = a \cdot sin(x) ) and ( y = a \cdot cos(x) ) where ( |a| ) is the amplitude.
- Period changes: ( y = sin(bx) ) and ( y = cos(bx) ), the period is ( \frac{2π}{|b|} ).
- Phase shifts and vertical shifts applied similarly.
Trigonometric Identities and Equations
- Basic Identities:
- Pythagorean identities:
- ( sin^2(θ) + cos^2(θ) = 1 )
- ( 1 + tan^2(θ) = sec^2(θ) )
- Angle Sum and Difference Identities:
- ( sin(a \pm b) = sin(a)cos(b) \pm cos(a)sin(b) )
- ( cos(a \pm b) = cos(a)cos(b) \mp sin(a)sin(b) )
- Solving Trigonometric Equations:
- Techniques include using identities, factoring, and inverse trigonometric functions.
4. Example Problems or Case Studies
Example Problem 1: Graphing a Sine Function
- Graph ( y = 2 \cdot sin(x) ).
- Solution:
- Amplitude is 2.
- The period is ( 2π ) (no change in b).
- Use key points: ( 0, \frac{π}{2}, π, \frac{3π}{2}, 2π ) to plot the graph.
Example Problem 2: Solving a Trigonometric Equation
- Solve ( sin(x) = \frac{1}{2} ) for ( 0 \leq x < 2π ).
- Solution:
- ( x = \frac{π}{6}, \frac{5π}{6} ).
5. Summary or Review Section
- Trigonometric functions express relationships between angles and sides of triangles.
- The unit circle is central for understanding sine, cosine, and tangent functions.
- Trigonometric identities simplify expressions and solve equations.
- Graphs of trig functions show their periodic nature and transformations.
6. Self-Assessment Questions
- Define the sine, cosine, and tangent functions using the unit circle.
- What is the amplitude of the function ( y = 3 \cdot cos(x) )?
- Graph ( y = sin(x + \frac{π}{2}) ).
- Solve for ( x ): ( 2cos(x) – 1 = 0 ).
- Explain the significance of the period and amplitude in the context of trigonometric functions.
7. Connections to Other Topics/Subjects
- Algebra: Using algebraic techniques to solve trigonometric equations.
- Physics: Application in wave motion, sound waves, and oscillations.
- Geometry: Using trigonometric functions to solve problems involving right and non-right triangles.
- Engineering: Trigonometric functions in signal processing and electrical engineering.
Encourage students to visually explore the unit circle and practice graphing using online tools or graphing calculators for deeper understanding.