Lesson 7-1 Objective: To learn the foundations of trigonometry.

Trigonometry The branch of mathematics that studies right triangles. It deals with the relationship between the sides and the angles. In order to do this you must also understand the relationship between angles and circles.

Angles An angle is formed by two rays with a common endpoint. (geometry definition) An angle is the result of a rotation of a ray about its endpoint. (trigonometry definition)

Standard Position When the initial side is on the positive x axis and the endpoint is on the origin Then the angle is in standard position.

Positive Angle – Standard Form α

Negative Angle – Standard Form β

Quadrant I Where the terminal side lays is where the angle is said to lie. Between 0 o and 90 o or 0 and

Quadrant II Between 90 o and 180 o or and

Quadrant III Between 180 o and 270 o or and

Quadrant IV Between 270 o and 360 o or and

Quadrantal When the terminal side lies on an axis it is called a quadrantal.

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 1o1o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 10 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 45 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 90 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 150 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 225 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 315 o

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 359 o

Review -Classifying Angles Acute Angle - measure greater than 0 degrees and less than 90 degrees

Review-Classifying Angles Obtuse - measure more than 90 degrees and less than 180 degrees (in Quadrant II )

Review - Classifying Angles Right - measure 90 degrees (Quadrantal)

23 Radian Measure To talk about trigonometric functions, it is helpful to move to a different system of angle measure, called radian measure. A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle.

©Carolyn C. Wheater, Radian Measure There are 2  radians in a full rotation - once around the circle (  is half a rotation) There are 360° in a full rotation (180 ° in half a rotation) To convert from degrees to radians or radians to degrees, use the proportion: degrees = radians 180°

©Carolyn C. Wheater, 2000 Sample Problems Find the degree measure equivalent of radians.  Find the radian measure equivalent of 210°

Radians (cont’d) Typically the angle is referred to as θ (theta) Some standard angle conversions radians = 180 o 1 radian =( ) o 1 o = radians

Examples Convert from degree to radian measure: -210 o 390 o

Examples Convert from radian measure to degrees. radians