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Exploring Trigonometric Functions and Angles: Practical Applications Guide

Learn angles, co-terminal angles, reference angles, trigonometric ratios, and solving real-world problems using trigonometry. Practice finding coterminal angles, reference angles, and trigonometric values. Enhance your understanding with practical examples.

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Exploring Trigonometric Functions and Angles: Practical Applications Guide

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  1. Chapter 5 The Trigonometric Functions Gettin’ Triggy wit’ it!

  2. Section 5.1 – Angles and Degree Measure Learning Targets: • I can find the number of degrees in a given number of rotations. • I can identify angles that are co-terminal with a given angle. • I can identify a reference angle for a given angle measure.

  3. Anatomy of an Angle Initial Side Terminal Side Vertex

  4. Standard Position

  5. Quadrantal angle

  6. Example 1: 5.5 x -360 -1980 3.3 x 360 1180 Give the angle measure represented by each rotation listed below: a) 5.5 rotations clockwise: b) 3.3 rotations counterclockwise:

  7. Definition 48 degrees 408 degrees Two angles in standard position are called coterminal angles if they have the same terminal side. Draw a pair of coterminal angles below.

  8. Example 2: 405o -315o -135o 585o Identify all angles that are coterminal with each angle. Then, find one positive angle and one negative angle that are coterminal with the angle. a) 45o b) 225o

  9. Example 3: 775 - 360 - 360 55o -1297 + 360 + 360 + 360 + 360 143o If each angle is in standard position, determine a coterminal angle that is between 0o and 360o. State the quadrant in which the terminal side lies. a) 775o b) -1297o

  10. Definition If a is a nonquadrantal angle in standard position, its reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis.

  11. Example 4: 60o 45o Find the measure of the reference angle for each angle. a) 120o b) -135o

  12. Warm up! 9.5 x -360 -3420 446, -274 86 + or - 360n -777/360 = -2.15.. 303 -777 + 720 = -57 48 15 CO-TERMINAL (Same ending position) “reference” can be used to simplify problems. 1. Find 9.5 rotations clockwise. 2. Find all angles coterminal to 86o. Then find one pos and neg. 3. If the angle -777o is in standard position, determine a coterminal angle that is between 0o and 360o. 4. Find the measure of the reference angle for each angle below: a) 312o b) -195o 5. What is the difference between coterminal angles and reference angles?

  13. 5.2 – Trig Functions Objectives: • Find the values of trigonometric ratios for acute angles of right triangles.

  14. Anatomy of a Triangle

  15. Example 1:

  16. Example 2:If cos x = ¾, find sec x.If sc x = 1.345, find sin x.

  17. b. If csc x = 1.345, find sin x.

  18. Start here with 2nd and 3rd hour

  19. 30-60-90 Triangle:45-45-90 Triangle: Pythagorean Thm Pythagorean Thm

  20. Example 3:Find x and y using the rules from the special right triangles above.

  21. Example 4:Find x and y using the rules from the special right triangles above.

  22. Warm-up Find x in the triangle below

  23. Hi.

  24. 30-60-90 Triangle:45-45-90 Triangle: Pythagorean Thm Pythagorean Thm

  25. Unit Circle 

  26. NOTE: stop for radians until after 6.1 mini lesson

  27. Section 6.1 (mini-lesson) Example 1: • Convert 330o to radian measure. • Change radians to degree measure. o

  28. NOTE: stop for radians until after 6.1 mini lesson

  29. 5.3 – Trig and the Unit Circle Objectives: • I can find the values of the six trig functions using the unit circle. • I can find the values of the six trig functions of an angle in standard position given a point on its terminal side.

  30. sin x = _______ csc x = ______ cos x = _______ sec x = ______ tan x = _______ cot x = ______ Example 1: • Using the unit circle, find the six trigonometric values for a 135 angle

  31. 6.1 (Mini-Lesson)

  32. Warm- Up Find the sin, cos and tan of (no need to write…. Just think)

  33. Tuesday – Watch video and do few examples below http://www.youtube.com/watch?v=X1E7I7_r3Cw

  34. Warm-Up Suppose x is an angle in standard position whose terminal side lies in Quadrant IV. If cos x =.5 , find the values of the remaining five trig functions of x. (Hint: Sketch a picture).

  35. Unit Circle time!!!!!!

  36. 5.4 – Word Problems and Trig Objectives: • Use trigonometry to find the measures of the sides of right triangles.

  37. Example 1:The circus has arrived and the roustabouts must put up the main tent in a field near town. A tab is located on the side of the tent 40 feet above the ground. A rope is tied to the tent at this point and then the rope is placed around a stake on the ground. a. If the angle that a rope makes with the level ground is 53, how long is the rope?Picture: Work:b. What is the distance between the bottom of the tent and the stake?

  38. Example 2: A regular pentagon is inscribed in a circle with diameter 8.34 centimeters. The apothem of a regular polygon is the measure of a line segment from the center of the polygon to the midpoint of one of its sides. Find the apothem for the pentagon.

  39. Definition: An angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level.Definition: An angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level.

  40. Example 3: (skip it)On May 18, 1980, Mt. Saint Helens erupted with such force that the top of the mountain was blown off. To determine the new height at the summit, a surveyor measured the angle of elevation to the top of the volcano to be 37. The surveyor then moved 1000 feet closer to the volcano and measured the angle of elevation to be 40. Determine the new height of Mt. Saint Helens.Picture: Work:

  41. YOU TRY IT!!The chair lift at a ski resort rises at an angle of 20.75 and attains a vertical height of 1200 feet. How far does the chair lift travel up the side of the mountain?

  42. You try it…answers…..Quiz tomorrow

  43. Warm-up The Ponce de Leon lighthouse in St. Augustine, FL, is the second tallest brick tower in the U.S. It was built in 1887 and rises 175 feet above sea level. How far from the shore is a motorboat if the angle of depression from the top of the lighthouse is 13 degrees?

  44. Ad Math students – please read • Mama ain’t happy. • It is clear to me that you have not been been checking on Moodle. The answers are numbered differently, but they are there. • Do not blame this on the snow days. • Many of you did well on the quiz. Many of you did not. • I am disappointed. • Come in. Grab your quiz. And immediately start figuring out what you did wrong. • DO.NOT. Do anything else. Period.

  45. 5.5 – Inverse Functions Objectives: • Evaluate inverse trig functions. • Find missing angle measurements. • Solve right triangles.

  46. Inverse trig functionsDiscussion:in x = ½ can be solved by using the Arcsin function:arcsin ½ = x which is read “x is the angle whose sine is ½ . How many solutions does arcsin ½ = x have?Similarly, there is an arccosine and arctangent function (arcos and arctan). We use these functions to ________.

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