FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.

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1 Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is part of a line that consists of two distinct points on the line and all the points between them. Line segment AB or segment AB, A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A. An angle is a plane figure formed by two rays that share a common point. Initial and terminal sides initial side Degree is a unit for measuring angles and arcs that corresponds to of a complete revolution. Counterclockwise rotation results in (+) measure Clockwise rotation results (-) measure Acute angle, Right angle, Obtuse angle Defns Complementary angles are angles whose sum measures that add up to 9 (they re complements of each other) Supplementary angles are angles whose sum measures that add up to 8 (they re supplements of each other) Portions of a degree are measured with minutes and seconds Defns One minute, written, is of a degree. = ( ) or = One second, written is of a minute. = ( ) = ( ) or = Ex 1 Find the measure of the smaller angle formed by the hands of a clock at the given time. a) b) Page 1 of 11

2 Ex 1 Find the measure of the smaller angle formed by the hands of a clock at the given time. c) 8 d) (#30) e) Challenge Ex 2 (#48) Perform the calculation. 9 8 Check your work. Ex 3 (#60) Convert the angle measure to degrees. If applicable, round to the nearest thousandth of a degree. 9 Ex 4 (#72) Convert the angle measure to degrees, minutes, and seconds. Round answer to the nearest second, if applicable. 8 Defns An angle is in standard position if its vertex is at the origin and its initial side is on the -axis. Angles in standard position whose terminal sides lie on the - or -axis, such as angles with 9 8, and so on, are called quadrant angles. Angles with the same initial and terminal sides, but different amounts of rotation are called coterminal angles. Their measures differ by a multiple of. Page 2 of 11

3 Ex 5 (#100) Give an expression that generates all angles coterminal with the angle. Let represent any integer. Find the angle with least positive measure coterminal with and. Ex 6 (#124) Locate the point ( ) and draw a ray from the origin to the point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance from the origin to the point, using the distance formula. Ex 7 (#134) An airplane propeller rotates 1000 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 1 sec. Sec 1.2: Angle Relationships and Similar Triangles Review (HW) Vertical angles (have same measure) Parallel lines Transversal Alternate interior angles (equal) Alternate exterior angles (equal) Interior angles on the same side of transversal (suppl.) Corresponding angles (equal) m n Fact The sum of the measure of the angles of any triangle is 8 Review (HW) Acute triangle Right triangle Obtuse triangle Equilateral triangle Isosceles triangle Scalene triangle Conditions for Similar Triangles For any triangle to be similar to triangle, the following conditions must hold. 1. Corresponding angles must have the same measure. 2. Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal.) Page 3 of 11

4 Ex 8 Find the measures of angles 1, 2, 3, and 4 in the figure, given that the lines and are parallel. 1 2 m (9x + 9) 4 3 ( x ) n Ex 9 Joey wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time that Joey, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree. Sec 3.1: Radian Measure Radian measure is a common unit of measure. In more theoretical work in math, radian measure is preferred as it allows us to treat the domain of trig functions as real numbers, rather than angles. It also simplifies theorems such as the derivative of the sine function. If you plan on moving on to Calculus, you must be able to work in radians. It was wide applications in engineering and science. Defn An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. radian think radius Exs Draw figures to represent = radians, = radian, and = radians. Page 4 of 11

5 Ex 10 What quadrant is = in? Ex 11 Convert each to degree or radian measure. a) b) c) d) 8 Converting Between Degrees and Radians = π 1. Deg Rad Multiply by π 8 2. Rad Deg Multiply by 8 π Note If no unit measure is specified, then the angle is understood to be measured in radians. Common Measures in Radians and Degrees Must know these equivalences Figure 4 on page 97 Degrees Radians 9 8 Print Unit Circle and/or Table (multiple timed quizzes on basic trig eval) Page 5 of 11

6 Sec 1.3: Trigonometric Functions We will define the six trig functions: sine, cosine, tangent, cosecant, secant, and cotangent. Let define an angle in standard position. Choose any point ( ) on the terminal side of. We define as follows: Six Trig Functions = = = = = = Where = + SOH-CAH-TOA Ex 12 The terminal side of an angle in standard position passes through the point (8 ). Find the values of the six trig functions of angle. Page 6 of 11

7 Ex 13 Find the six trig function values of the angle in standard position, if the terminal side of is defined by =. Ex 14 Graph =. Use the graph to determine what tangent really means. Quadrantal Angles Formed when terminal side of an angle in standard position lies along one of the axes. Occurs when { } Trig Function Values at Quadrantal Angles in degrees in radians Page 7 of 11

8 Ex 15 (#92) Evaluate. ( 8 ) + ( 8 ) Note: (Pyth Id) Ex 16 Decide whether each expressions is equal to 0, 1, or -1 or is undefined. (#94) (#96) (#100) PP (#102) PP [ ] [( + ) ] [ ] [ ] Ex 17 Evaluate. ( 9 ) + Note that = = = Sec 1.4: Using the Definitions of the Trig Functions Reciprocal Identities = = = = = = Ex 18 Find each function value. a) b) c) (#10) = = = Page 8 of 11

9 Signs of Function Values ALL STUDENTS TAKE CALCULUS Ex 19 Determine the signs of the trig functions of an angle in standard position with the given measure. a) b) c) d) 8 8 Ex 20 Identify the quadrant (or possible quadrants) of an angle that satisfies the given conditions. a) b) c) (#36) Find the Ranges of Trig Functions Page 9 of 11

10 Ranges of Trig Functions Ex 21 Decide whether each statement is possible. a) b) c) d) = 999 = = = 8 Ex 22 Suppose that is in QIII and =. Find the values of the other 5 trig functions. Derive Pythagorean Identities Pythagorean Identities For all angles for which the function values are defined, the following identities hold. + = + = + = Quotient Identities For all angles for which the denominators are not zero, the following identities hold. = = Page 10 of 11

11 Ex 23 Find and, given that = and. Check your answer. Use identity. Ex 24 Find and, given that = and is in QII. Ex 25 (#74) Find the remaining 5 trig function values if = and is in QII. Page 11 of 11

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