Trigonometric Ratios and Functions notes student.notebook August 21, 2013

Transcription

1 Trigonometric Ratios and Functions title page 1

2 triangles II triangles basic trigonometry basic trigonometry II 2

3 triangles II triangles basic trigonometry basic trigonometry II 3

4 Find all missing sides triangles II triangles basic trigonometry basic trigonometry II 4

5 Review: A Pythagorean Thm b c C a B triangles II triangles basic trigonometry basic trigonometry II 5

6 Find the missing side Pythagorean theorem 6

7 Right Triangle Trigonometry Six trig functions: opposite side hypotenuse sine cosine tangent cosecant secant cotangent adjacent side triangles II triangles basic trigonometry basic trigonometry II 7

8 The abbreviations opp, adj, and hyp represent the lengths of the three sides of the right triangle. Note: csc, sec and cot are the reciprocals of sin, cos and tan respectively triangles II triangles basic trigonometry basic trigonometry II 8

9 Find the missing side and then evaluate the six trigonometric functions of the angle triangles II triangles basic trigonometry basic trigonometry II 9

10 Solve ABC using the diagram and the given measurements A b c C a B triangles II triangles basic trigonometry basic trigonometry II 10

11 To measure the width of a river you plant a stake on one side of the river, directly across from a boulder. You then walk 100 meters to the right of the stake and measure a angle between the stake and the boulder. What is the width of the river triangles II triangles basic trigonometry basic trigonometry II 11

12 You are climbing Mount Cook in New Zealand. You are below the mountain's peak at an altitude of 8580 feet. Using surveying instruments, you measure the angle of elevation to the peak to be. The distance (along the face of the mountain) between you and the peak is 7426 feet. What is the altitude of the peak? angle of depression angle of elevation triangles II triangles basic trigonometry basic trigonometry II 12

13 General Angles and Radian Measure (13.2) Radian Measure - Activity 1. Use a compass and draw a large circle on a piece of paper. 2. Identify the circle (O) and draw a radius horizontally from O toward the right, intersecting the circle at point A. 3. Cut a piece of string the same size as the radius. 4. Place one end of the string at A and bend it around the circle counter clockwise, marking the point B on the circle where the other end of the string ends up. 5. Draw the radius from O to B. The measure of is one radian! General Angles and Radian Measure 13

14 Questions/procedures continued (bonus possibly...) 1. Label a point C on the circle where the is 2 radians. 2. Label a point D on the circle where the is 3 radians. 3. Label a point E on the circle where the is 4 radians. 4. Label a point F on the circle where the is 5 radians. 5. Label a point G on the circle where the is 6 radians. General Angles and Radian Measure 14

15 Questions: 1. What is the circumference of the circle, in terms of its radius, r? 2. How many radians must there be in a complete circle? 3. If we cut a piece of string 3 times as big as the radius, would it extend halfway around the circle? Why or why not? 4. How many radians are in a straight angle? General Angles and Radian Measure 15

16 One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. Because the circumference of a circle is radians in a full circle., there are General Angles and Radian Measure 16

17 Unit circle 17

18 Understanding Unit circle Points unit circle 18

19 The angles,, and occur frequently in trigonometry. The table below gives the values of the six trionometric functions for these angles triangles II triangles basic trigonometry basic trigonometry II 19

20 Angles in Standard Position You can generate any angle by fixing one ray, called the, and rotating the other ray, called the, about the vertex. In a coordinate plane, an angle whose vertex is at the origin and whose initial side is the positive x-axis is in. The measure of an angle is determined by the amount and direction of rotation from the initial side to the terminal side. The angle measure is positive if the rotation is counter-clockwise and negative if the rotation is clockwise. The terminal side of an angle can make more than one complete rotation. Angles in standard Position 20

21 Draw an angle with the given measure in standard position. Angles in standard Position 21

22 Find one positive and one negative angle coterminal with the given angle. (add/subtract or multiples of ) Coterminal angles 22

23 Conversions Between Degrees and Radians To rewrite a degree measure in radians, multiply by To rewrite a radian measure in degrees, multiply by Coversions between degrees and radians 23

24 Rewrite each degree measure in radians and each radian measure in degrees. conversions between degrees and radians 24

25 Arc lengths and Areas of Sectors A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle radii. of a sector is the angle formed by the two Arc length: Area: central angle sector arc length s Arc length and Area of a Sector 25

26 Examples: (remember arc length and area or to be in radians) Find the arc length and area of a sector with the given radius r and central angle Arc length and Area of a Sector 26

27 Trigonometric Functions of Any Angle General Definition of Trigonometric Functions Let be an angle in standard position and (x,y) be any point (except the origin) on the terminal side of. The six trigonometric functions of are defined as follows: r Pythagorean theorem gives: Trig functions of any angle 27

28 Evaluating Trigonometric Functions Given a Point Example: Let (3,-4) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of. r (3,-4) Trig functions of any angle - given a point 28

29 Use the given point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of. Trig functions of any angle-given a point 29

30 If the terminal side of lies on an axis, then is a quadrantal angle. Evaluate the six trionometric functions of Trig functions of any angle 30

31 The values of trigonometric functions of angles greater than (or less than ) can be found using corresponding acute angles called reference angles. Let be an angle in standard position. Its reference angle is the acute angle (read theta prime) formed by the terminal side of and the x-axis. Trig functions of any angle 31

32 Sketch the angle. Then find its reference angle. Trig functions of any angle 32

33 Evaluating Trigonometric Functions Steps: 1. Find the reference angle 2. Evaluate the trig function for the angle 3. Use the quadrant in which lies to determine the sign of the trig function value of Evaluating Trig functions 33

34 The angles,, and occur frequently in trigonometry. The table below gives the values of the six trionometric functions for these angles triangles II triangles basic trigonometry basic trigonometry II 34

35 Evaluate the function without using a calculator. Evaluating trig functions 35

36 Use a calculator to evaluate the function. Round the result to four decimal places. evaluating trig functions 36

37 Calculating Projectile Distance The horizontal distance (in feet) traveled by a projectile with an initial speed (in feet per second) is given by where is the angle at which the projectile is launched. Gina is playing soccer with her friend. They both kick the ball with an initial speed of 24 feet per second. Gina's kick projected at an angle of and her friend's kick was projected at an angle of. About how much farther will Gina's soccer ball go than her friend's. calculating projectile distance 37

38 13.4 Inverse Trigonometric Functions (Finding angles that correspond to a given value of a trigonometric function.) If then the of is where and If then the of is where and If then the of is where and Inverse Trig Functions 38

39 Evaluating Inverse Trigonometric Functions. Evaluate the expression in both radians and degrees Hint: (look at chart) -(use calculator in degrees - change to radians) Inverse Trig Functions 39

40 Finding an angle measure Find the measure of the angle for the triangle shown Round to three significant digits Inverse Trig Functions 40

41 Use a calculator to evaluate the expression in both radians and degrees. Round to three significant digits. Inverse Trig Functions 41

42 Solving a trigonometric equation Solve the equation for. Round to three significant digits. Inverse Trig Functions 42

43 The swimming pool shown in cross section ranges in depth from 3 feet at the shallow end to 8 feet at the deep end. Find the angle of depression between the shallow end and the deep end. 30 ft 3 ft 10 ft 8 ft 10 ft Inverse Trig Functions 43

44 The Cherry Street Bridge in Toledo Ohio, is a double-leaf (or bascule)drawbridge. Each leaf of the bridge is approximately 200 feet long. A ship that is 150 feet wide needs to pass through the bridge. What is the minimum angle that each leaf of the bridge should be opened to in order to ensure that the ship will fit? 150 ft 200 ft 200 ft Inverse Trig Functions 44