Review of College Algebra and Geometry

Square Roots and Properties

Understanding square roots and their properties is essential for solving equations and working with geometric figures in trigonometry.

• Square Root Definition: For a positive number x, is the positive number whose square is x.

• Undefined for Negatives: is not defined for x < 0 in the set of real numbers.

• Examples: , ,

• Product Rule: for

• Quotient Rule: for

Parallel Lines: Lines in the same plane that never intersect. Perpendicular Lines: Lines that intersect at a right angle (90°).

Angle: Formed by two rays with a common endpoint.

Similar and Congruent Shapes

Understanding similarity and congruence is foundational for trigonometry, especially when working with triangles.

• Similar Shapes: Same shape, corresponding angles are equal, and sides are proportional.

• Congruent Shapes: Same shape and size; corresponding sides and angles are equal.

Proof: A logical argument that uses deductive reasoning to show a statement is true.

Measuring Angles and Circles

Area and Circumference of Circles

Circles are fundamental in trigonometry, especially for defining angles and trigonometric functions.

• Area of a Circle: , where r is the radius.

• Circumference of a Circle:

• Diameter:

Example: If the circumference of a circle is , then .

Sectors and Arcs

A sector is a region of a circle bounded by two radii and the arc between them. The arc is a portion of the circle's circumference.

• Area of a Sector: , where is the central angle in degrees.

• Arc Length:

Example: For a circle of radius 4 and a central angle of 60°, the area of the sector is , and the arc length is .

Central and Inscribed Angles

Angles in circles are classified based on their vertex location.

• Central Angle: Vertex at the center of the circle; its measure equals the intercepted arc.

• Inscribed Angle: Vertex on the circle; its measure is half the intercepted arc.

Example: If an inscribed angle intercepts an arc of 80°, the angle measures 40°.

Triangles and Polygons

Triangles: Types, Area, and Perimeter

Triangles are the basis for trigonometric ratios and identities.

• Area of a Triangle:

• Perimeter: Sum of all side lengths.

• Equilateral Triangle: All sides and angles are equal.

Example: For a triangle with sides 3, 4, and 5, and perimeter .

Rectangles and Squares

Rectangles and squares are special quadrilaterals with right angles, often used in geometric proofs and area calculations.

• Area of a Rectangle:

• Perimeter of a Rectangle:

• Area of a Square:

• Perimeter of a Square:

Surface Area and Volume of Solids

Prisms, Cylinders, and Spheres

Understanding three-dimensional shapes is important for applications of trigonometry in geometry and physics.

• Rectangular Prism:

  • Surface Area:

  • Volume:

• Cylinder:

  • Surface Area:

  • Lateral Area:

  • Volume:

• Sphere:

  • Surface Area:

  • Volume:

Right Triangles and the Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a fundamental result relating the sides of a right triangle, forming the basis for trigonometric ratios.

• Theorem Statement: In a right triangle with legs a and b, and hypotenuse c:

• Example: If and , then .

Additional info:

Some content, such as basic area and volume formulas, is foundational for trigonometry but may also be covered in geometry courses. These notes provide a comprehensive review relevant for students beginning a college trigonometry course.