- Convert angle measures between degrees and radians
- Interpret and apply basic trigonometric definitions and identities
- Analyze trigonometric functions by identifying graphs and periods, and describing shifts in sine or cosine graphs.
Degrees versus Radians
The Main Idea
- Degrees:
- One degree is of a full circular rotation
- Full circle =
- Symbol: (e.g., )
- Radians:
- Based on the radius of a circle
- One radian is the angle subtended by an arc length equal to the radius
- Full circle = radians
- Symbol: rad (e.g., rad)
- Conversion:
- radians
- radian
- Conversion formulas:
- Degrees to Radians:
- Radians to Degrees:
- Usage:
- Degrees: common in everyday situations, navigation, construction
- Radians: preferred in advanced mathematics, physics, engineering
- Express using radians.
- Express rad using degrees.
The Six Basic Trigonometric Functions
The Main Idea
- The Six Functions:
- Sine ():
- Cosine ():
- Tangent ():
- Cosecant (): (reciprocal of sine)
- Secant (): (reciprocal of cosine)
- Cotangent (): (reciprocal of tangent)
- Mnemonic Device:
- SOH-CAH-TOA for sine, cosine, and tangent
- Reciprocal Relationships:
- Applications:
- Finding unknown side lengths in right triangles
- Solving real-world problems involving angles and distances

A house painter wants to lean a -ft ladder against a house. If the angle between the base of the ladder and the ground is to be , how far from the house should she place the base of the ladder?
Trigonometric Identities
The Main Idea
- Trigonometric identities are equations involving trigonometric functions that are true for all values where they are defined.
- Types of Identities:
- Reciprocal Identities (e.g., )
- Pythagorean Identities (e.g., )
- Addition and Subtraction Formulas
- Double-Angle Formulas
- Importance:
- Crucial for solving trigonometric equations
- Used in proving other mathematical statements
- Frequently applied in calculus and higher mathematics
- Verification Methods:
- Manipulate one side of the equation to match the other
- Use known identities to make substitutions
- Convert all terms to sines and cosines if needed
Prove the trigonometric identity .
Simplify the expression by rewriting and using identities:
Graphs and Periods of the Trigonometric Functions
The Main Idea
- Periodicity:
- Trigonometric functions repeat their values at regular intervals
- Sine, cosine, secant, cosecant: period of
- Tangent, cotangent: period of
- Basic Graphs:
- Sine and cosine: smooth wave patterns
- Tangent and cotangent: repeating vertical asymptotes
- Secant and cosecant: repeating reciprocal patterns of cosine and sine
- Transformations: General form: (similar for cosine)
- : Amplitude (vertical stretch/compression)
- : Frequency (horizontal stretch/compression, affects period)
- : Phase shift (horizontal shift)
- : Vertical shift
- Effects of Transformations:
- Amplitude: is the height from midline to peak/trough
- Period: for sine and cosine
- Phase Shift: Right by if positive, left if negative
- Vertical Shift: Up by if positive, down if negative
Describe and sketch the graph of
π radians is equal to 180∘.