6.1 An Introduction to Angles

6.1.1 Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. They can be found by adding or subtracting multiples of (or radians) to a given angle.

• Definition: Two angles are coterminal if their difference is a multiple of or radians.

• Example: and are coterminal because .

• How to find coterminal angles: Add or subtract (or radians) as needed.

6.1.2 Degrees and Radians

Degrees and radians are two units for measuring angles. Converting between them is essential in trigonometry.

• Conversion formulas:

  • Degrees to radians:

  • Radians to degrees:

• Example: radians; radians.

6.2 Applications of Radian Measure

6.2.1 Area of a Sector

The area of a sector of a circle can be found using the central angle in radians.

• Formula: (where is in radians)

• Example: Find the area of a sector with radius 8 cm and central angle (convert to radians first).

6.2.2 Arc Length

The length of an arc intercepted by a central angle in a circle is proportional to the radius and the angle in radians.

• Formula: (where is in radians)

• Example: Find the arc length for a central angle of in a circle of radius 35 m.

6.2.3 Angular and Linear Velocity

Angular velocity measures how fast an angle changes, while linear velocity measures how fast a point moves along a circular path.

• Formulas:

  • Angular velocity:

  • Linear velocity:

• Example: Find the linear velocity in miles per hour of a point on the tip of a blade with a diameter of 10 inches, given the blade rotates at a certain rate.

6.3 Right Triangle Trigonometry

6.3.1 Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle.

• Formula: (where is the hypotenuse)

• Example: If and , find .

6.3.2 Applications

• Use the Pythagorean Theorem to solve right triangle problems, such as finding the length of a rectangle's diagonal.

• Example: If the diagonal is 2 inches more than the length, find the rectangle's dimensions.

6.4 Right Triangle Trigonometry

6.4.1 Right Triangle Setup

• Label the sides and angles of a right triangle and use trigonometric ratios to solve for unknowns.

• Example: Suppose , , and . Find , , and .

6.4.2 Cofunction Identities

• Cofunction identities relate the trigonometric functions of complementary angles.

• Example:

6.5 Trigonometric Functions of General Angles

6.5.1 Terminal Ray Approach

• Find trigonometric function values using a point on the terminal ray.

• Example: Find the trig function values for on the terminal ray.

6.5.2 Reference Angles

• Reference angles are the acute angles formed by the terminal side of an angle and the x-axis.

• Example: What is the reference angle for ? For radians?

6.6 The Unit Circle

6.6.1 Unit Circle Values

• Use the unit circle to find exact trigonometric function values for common angles.

• Example: Find the exact trig function values for .

6.6.2 Pythagorean Identities

• Know the three Pythagorean identities:

• Example: Use to find if and is in quadrant IV.

7.1 & 7.2 The Graphs of Sine and Cosine

7.1.1 Graphs of Sine and Cosine

• Graph transformations of sine and cosine, including reflection, amplitude, period, phase shift, and vertical shift.

• Example: Graph the first period of .

• Example: Graph the first period of .

7.3 The Graphs of Tangent, Cotangent, Secant, and Cosecant

7.3.1 Graphs of Tangent, Cotangent, Secant, and Cosecant

• Graph transformations of these functions, identifying reflection, amplitude, period, phase shift, and vertical shift.

• Example: Graph the first period of .

• Example: Graph the first period of .

7.3.2 Guide Function for Secant and Cosecant

• Know how to find and use the guide function for secant and cosecant, which is necessary for graphing these functions.

7.3.3 Domain and Range

• Be able to find the domain and range for all six trigonometric functions.

7.4 Inverse Trigonometric Functions

7.4.1 Inverse Trig Functions

• Find and evaluate inverse trigonometric functions.

• Example: , , , , , , .

7.4.2 Range of Inverse Trig Functions

• Identify the range of each inverse trigonometric function.

• Example: What is the range of ? ? ?

7.5 Inverse Trigonometric Functions

7.5.1 Compositions of Trig and Inverse Trig Functions

• Evaluate compositions of trigonometric and inverse trigonometric functions.

• Example: , , .

• Look for patterns in these compositions.

Summary Table: Trigonometric Functions and Their Properties

Additional info: This study guide covers the essential trigonometric concepts, identities, and graphing techniques required for a Precalculus course, including applications of radian measure, the unit circle, and inverse trigonometric functions.