1. An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.

2. An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise rotation.

3. An angle is called a quadrantal angle if its terminal side lies on the x-axis or on the y-axis. Angle is an example of a quadrantal angle. Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360°. An acute angle measures less than 90°. A right angle measures 90°. An obtuse angle measures more than 90° but less than 180°. A straight angle measures 180°.

4. Complementary Angles: Supplementary Angles: Find the complement and supplement angles of 40 o. The sum of any two angles that equals 90 o. The sum of any two angles that equals 180 o. 90 o – 40 o = 50 o 180 o – 40 o = 140 o means 90 o

5. There are 60 minutes in 1 degree. There are 60 seconds in 1 minute. 75 min. > 60 min., so carry 1 degree. We need to borrow 1 degree and convert it to 60 min.

6. Convert the minutes and seconds to fractional degrees. Calculator help! Round your answer according to the directions. The whole number is the degrees. Multiply the decimal by 60 to determine the minutes. The whole number is the minutes. Multiply the decimal by 60 to determine the seconds.

7. Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles. Increasing or decreasing the angle measure of an angle in standard position by an integer multiple of 360 o results in a coterminal angle. Thus, an angle of is coterminal with angles of, where k is an integer. Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following: a. a 400° angle b. a –855° angle

8. CD players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through how many degrees will a point on the edge of the CD move in 2 seconds? Determine how many revolutions in 1 second. 8 revolutions in 1 second times 2 is 16 revolutions. 16 revolutions times 360 o.

9. In the diagram, to the right, 2 parallel lines are intersected by a transversal line. Are formed by 2 intersecting lines and are the angles that open opposite of each other. Name all the pairs of vertical angles.1 & 4,2 & 3,5 & 8,and 6 & 7. Are the matching angles formed by the two intersections of the parallel lines. Name all the pairs of corresponding angles.1 & 5,2 & 6,3 & 7,and 4 & 8. Vertical angles are always equal in measure. Corresponding angles are always equal in measure. Are the angles in between the parallel lines and alternate over the transversal line. Symbol for parallel. Name all the pairs of alternate interior angles.3 & 6and 4 & 5. Alternate interior angles are always equal in measure. Are the angles outside the parallel lines and alternate over the transversal line. Name all the pairs of alternate exterior angles.1 & 8and 2 & 7. Alternate exterior angles are always equal in measure. Are the angles in between the parallel lines and on the same side of the transversal line. Name all the pairs of same side interior angles.3 & 5and 4 & 6. Same side interior angles are always supplementary.

10. Sum of the angles of a triangle_______________________________.always adds up to 180 degrees All three angles are acute, less than 90 o. One angle is 90 o. One angle is greater than 90 o.

11. All three sides are equal. Two sides are equal. No sides are equal. Two triangles that are the same shape, but not necessarily the same size. 1. Corresponding angles must be the same measure. 2. Corresponding sides must be proportional. 1.2.

12. Examples. Find the measure of all angles. Find the measure of all angles and sides. ABC ~ RST

13. Definitions of Trigonometric Functions of Any Angle. Let be any angle in standard position and let P = (x, y) be a point on the terminal side of. If is the distance from (0, 0) to (x, y), the six trigonometric functions of are defined by the following ratios: What do you notice about the fractions?Reciprocals! Pythagorean Theorem

14. Let P = (8, 15) be a point on the terminal side of. Find each of the six trigonometric functions of. P = (8, 15) is a point on the terminal side of. x = 8 and y = 15.

15. Let P = (1, –3) be a point on the terminal side of. Find each of the six trigonometric functions of. P = (1, –3) is a point on the terminal side of. x = 1 and y = –3.

16. Find the six trigonometric function values of the angle in standard position, if the terminal side of is defined by 3x + 4y = 0, x < 0. x < 0 means that the terminal side is in quadrant 2 or 3! Solve the linear equation for so it is in slope intercept form, y = mx + b. Start at (0, 0) and slope of -3/4. Since we are in quadrant 2 or 3, we will have to reverse the slope…up 3, left 4.

17. Section 1.4Reciprocal Identities. Flip the fractions Use this same concept for the other trig. functions we get the rest of the identities

18. We know that r is always positive and x < r & y < r. sin & cscSin & csc sin & csc cos & sec Cos & sec cos & sec tan & cot Tan & cot tan & cot A All S Students T Take C Calculus Since x and y are -1. Since x and y can be all real numbers, dividing real numbers will still be real numbers. Since x and y are 1. If x and y are negative, the fraction < -1.

19. Use All Students Take Calculus Negative AS TC ++ – – Quadrant 2 has the sign representation for both conditions, therefore, the terminal side is in quadrant 2. Positive Use the Pythagorean Theorem. x has to be negative because we are in quadrant 2.

20. Divide by r 2 Divide by y 2 Divide by x 2

21. AS TC + + – – NegativePositive Since we flipped the sine and cosine, the reciprocal of tangent is cotangent. Positive, Quad. 2 Using identities may be quicker. Pythagorean and Quotient.

22. AS T C Negative Positive Negative – 3 – 4 Using Pythagorean & Reciprocal Identities.