Trigonometric Functions and Relationships

Finding All Six Trigonometric Functions

Given the value of one trigonometric function for an angle, you can determine the remaining five using fundamental identities and relationships.

• Key Point: The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.

• Key Point: Use the Pythagorean identity to find missing values.

• Example: If and is in Quadrant II, find :

Solving Trigonometric Equations

Solving on a Given Interval

Trigonometric equations can be solved for all solutions within a specified interval, often or .

• Key Point: Isolate the trigonometric function and use inverse functions to solve.

• Example: Solve for in :

Right Triangle Applications

Solving Word Problems and DMS Conversion

Right triangle problems often require converting between degrees-minutes-seconds (DMS) and decimal degrees, and applying trigonometric ratios.

• Key Point: Use , , and to relate sides and angles.

• Key Point: DMS to decimal:

• Example: Convert to decimal:

Law of Cosines

Solving Triangles

The Law of Cosines is used when you know two sides and the included angle, or all three sides of a triangle.

• Key Point:

• Example: Given , , , find :

Inverse Trigonometric Functions

Finding Exact Values

Inverse trigonometric functions return the angle whose trigonometric function equals a given value.

• Key Point: , , are principal values.

• Example:

Simplifying Trigonometric Expressions

Algebraic Techniques

Algebraic methods such as factoring, finding least common denominators (LCDs), using conjugates, and FOIL are used to simplify trigonometric expressions.

• Key Point: Factor expressions and use identities to simplify.

• Example: Simplify :

  • , so

Proving Trigonometric Identities

Providing Reasons

To prove a trigonometric identity, transform one side using known identities and algebraic steps, justifying each step.

• Key Point: Use fundamental identities: Pythagorean, reciprocal, quotient, and cofunction identities.

• Example: Prove using quotient identity.

Sum and Difference Identities

Finding Equivalent Expressions

Sum and difference identities allow you to express trigonometric functions of sums or differences of angles.

• Key Point:

• Example:

Parametric Equations

Converting to Rectangular Form

Parametric equations express and in terms of a parameter . To convert to rectangular form, eliminate $t$.

• Key Point: Solve for in one equation and substitute into the other.

• Example: , ; , so

Complex Numbers and Trigonometric Form

Converting Between Forms

Complex numbers can be written in rectangular form () or trigonometric form ().

• Key Point: ,

• Example: ; ,

Polar and Rectangular Coordinates

Converting Between Systems

Points can be represented in polar coordinates or rectangular coordinates .

• Key Point: ,

• Key Point: ,

• Example: Convert to rectangular: ,

Law of Sines

Solving Word Problems

The Law of Sines is used when you know two angles and one side, or two sides and a non-included angle.

• Key Point:

• Example: Given , , , find :

Right Triangles and Bearings

Solving Word Problems

Bearing problems involve direction and angle measurements, often requiring right triangle trigonometry.

• Key Point: Draw a diagram and use , , or as appropriate.

• Example: A ship sails 10 miles north, then 10 miles east. Find the bearing from the starting point.

Graphing Sine and Cosine Functions

Amplitude, Period, and Phase Shift

The general form of a sine or cosine function is .

• Key Point: Amplitude is ; Period is ; Phase Shift is

• Example: : Amplitude , Period , Phase Shift

Solving Trigonometric Equations (All Solutions)

Finding All Solutions

Trigonometric equations may have infinitely many solutions; general solutions are expressed using periodicity.

• Key Point: or

• Example: : ,

Coterminal and Reference Angles

Finding Coterminal and Reference Angles

Coterminal angles share the same terminal side; reference angles are the acute angle formed with the x-axis.

• Key Point: Coterminal: Add or subtract or

• Key Point: Reference angle: For in Quadrant II, reference angle

• Example: coterminal with ; reference angle for is

Reference Triangles

Drawing and Labeling Reference Triangles

Reference triangles are drawn for any angle in standard position, with sides labeled according to trigonometric ratios.

• Key Point: The reference angle is always positive and less than .

• Example: For , reference angle is ; triangle sides labeled as opposite, adjacent, hypotenuse.

Linear and Angular Speed

Solving Word Problems

Linear speed is the rate of change of distance; angular speed is the rate of change of angle.

• Key Point: , where is linear speed, is radius, is angular speed.

• Example: A wheel of radius 2 m rotates at $3v = 2 \times 3 = 6$ m/s.

Summary Table: Trigonometric Functions and Relationships

Additional info: These notes cover all major trigonometry and analytic geometry topics listed for a precalculus final exam, including algebraic manipulation, solving equations, and applications. Students should review old exams, homework, and quizzes for practice.