8.1 Trigonometric Identities

Odd/Even Properties

Trigonometric functions exhibit specific symmetry properties, classified as odd or even functions. Understanding these properties helps simplify expressions and solve equations.

• Odd Functions: Satisfy . Examples: , .

• Even Functions: Satisfy . Example: .

Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined.

• Example: Verify .

• Example: Verify .

8.2 The Sum and Difference Formulas

Sum and Difference Formulas

These formulas allow you to find the sine, cosine, and tangent of sums and differences of angles.

• Sine:

• Cosine:

• Tangent:

Application: Find exact values for angles such as , , , , , , (in radians).

Double-Angle Formulas

These formulas express trigonometric functions of double angles in terms of single angles.

Example: Find , , and if and is in Quadrant IV.

Half-Angle Formulas

These formulas allow you to find the sine, cosine, and tangent of half an angle.

Example: Find using a half-angle formula.

8.5 Trigonometric Equations

Solving Trig Equations

Solving trigonometric equations involves finding all solutions within a specified interval, often or .

• Find all radian solutions for in .

• Find all radian solutions for in .

• Find all radian solutions for in .

• Find all degree solutions for and round your answers to the nearest tenth of a degree.

9.1 Right Triangle Applications

Right Triangle Applications

These problems involve applying trigonometric ratios to solve for unknown sides or angles in right triangles, often in real-world contexts.

• Angle of Elevation: The angle formed by the line of sight above the horizontal.

• Angle of Depression: The angle formed by the line of sight below the horizontal.

• Example: A minute hand is 25.5" and a circular entrance has been carved into the side of a vertical cliff. From a distance of 100 feet from the base of the cliff, the angle of elevation to the bottom is and the angle of elevation to the top is . Find the height of the entrance.

9.2 The Law of Sines

Oblique Triangles with One Solution

The Law of Sines is used to solve for unknown sides or angles in non-right (oblique) triangles when given sufficient information.

• Example: , ,

Oblique Triangles with No Solution

Sometimes, the given information leads to no possible triangle (e.g., the side is too short for the given angle).

• Example: , ,

9.3 The Law of Cosines

SAS and SSS Oblique Triangles

The Law of Cosines is used to solve for unknown sides or angles in oblique triangles when two sides and the included angle (SAS) or all three sides (SSS) are known.

• Example (SAS): , ,

• Example (SSS): , ,

10.1 Polar Coordinates and Polar Equations

Polar to Rectangular; Rectangular to Polar

Polar coordinates represent points in the plane using a radius and angle. Conversion between polar and rectangular forms is essential for graphing and solving equations.

• Polar to Rectangular: ,

• Rectangular to Polar: ,

• Example: Convert to rectangular form.

• Example: Convert to polar form.

10.3 Complex Numbers in Polar Form; DeMoivre's Theorem

Polar Form of Complex Numbers

A complex number can be represented in polar form as , where is the modulus and is the argument.

• Conversion: ,

• Example: Convert into polar form.

• Example: Convert into rectangular form.

Multiplication and Division

Multiplying and dividing complex numbers in polar form is simplified using their moduli and arguments.

• Multiplication:

• Division:

• Example: Find and for and .

DeMoivre's Theorem

DeMoivre's Theorem provides a formula for raising complex numbers in polar form to integer powers:

• Example: Evaluate and put your answer into rectangular form.

Roots of a Complex Number

The th roots of a complex number can be found using the formula:

• , for

• Example: Find the fourth roots of .

• Example: Find the third roots of .

• Example: Find the fourth roots of .