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

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.

Anatomy of an Angle Initial Side Terminal Side Vertex

Standard Position

Quadrantal angle

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

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

Example 2: 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 405o -315o -135o 585o

Example 3: 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 775 - 360 - 360 55o -1297 + 360 + 360 + 360 + 360 143o

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.

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

Warm up! 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? 9.5 x -360 -3420 86 + or - 360n 446, -274 -777/360 = -2.15.. 303 -777 + 720 = -57 48 15 CO-TERMINAL (Same ending position) “reference” can be used to simplify problems.

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

Anatomy of a Triangle

Example 1:

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

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

Start here with 2nd and 3rd hour

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

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

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

Warm-up Find x in the triangle below

Hi.

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

Unit Circle 

NOTE: stop for radians until after 6.1 mini lesson

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

NOTE: stop for radians until after 6.1 mini lesson

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.

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

6.1 (Mini-Lesson)

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

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

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).

Unit Circle time!!!!!!

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

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?

Example 2: A regular pentagon is inscribed in a circle with diameter 8 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.

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.

Example 3: (skip it) On May 18, 1980, Mt 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:

YOU TRY IT. The chair lift at a ski resort rises at an angle of 20 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?

You try it…answers….. Quiz tomorrow

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?

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.

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

Inverse trig functions Discussion: 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 ________.

Example 1: Solve each equation: a. b. c.

Example 2: Evaluate each expression Example 2: Evaluate each expression. Assume that all angles are in Quadrant I. a. b.

Example 3: If f = 17 and d = 32, find E when D is a right angle.

Example 4: (skip) A security light is being installed outside a loading dock. The light is mounted 20 feet above the ground. The light must be placed at an angle so that it will illuminate the end of the parking lot. If the end of the parking lot is 100 feet from the loading dock, what should be the angle of depression of the light?

Example 5: Solve the triangle (find all missing sides and angles): A = 33, b = 5.8, and C = 90 a = 23, c = 45, and C = 90o

Warm-up Solve tan(cos-1 ) if theta is in the 4th Quad cos(cos-1 ) if theta is in the 1st quad sin(tan-1 ) if theta is in the second quad

5.6 Objectives: Solve triangles using the Law of Sines Find the area of a triangle if the measures of two sides and the included angle or the measures of two angles and a side are given.

Law of Sines: The Law of Sines can be used to solve triangles that are NOT right triangles. When do you use Law of Sines? ASA and AAS triangles!

Example 1: Solve the triangle when A = 33, B = 105, and b = 37.9.

Example 2: A baseball fan is sitting directly behind home plate in the last row of the upper deck of US Cellular Field in Chicago. The angle of depression to home plate is 29.75o and the angle of depression to the pitcher’s mound is 24.25o. In major league baseball, the distance between home plate and the pitcher’s mound is 60.5 feet. How far is the fan from home plate?

Area of a Triangle The area of a triangle is ½ bh Area of a Triangle The area of a triangle is ½ bh. By manipulating the formula using the Law of Sines, we get a new area formula: Area = ½ bc sin A Example 3: Find the area of the triangle with a = 4.7, c = 12.4, and B = 47.

Warm up The center of the Pentagon in Arlington, Virginia, is a courtyard in the shape of a regular pentagon. The courtyard could be inscribed in a circle with radius of 300 ft. Find the area of the courtyard. (WORK TOGETHER to break this apart!!!!)

Typo on Area problem #6 and #7

5.8 Objectives: Solve triangles using the Law of Cosines Find the area of a triangle if the measures of three sides are given.

Law of Cosines: The Law of Cosines can be used to solve triangles that are NOT right triangles. When do you use Law of Cosines? SSS and SAS triangles!

Example 1: Solve the triangle when A = 120, b = 9, and c = 5.

Example 2: Solve the triangle when a = 24, b = 40 and c = 18.

Example 3: For a right-handed golfer, a slice is a shot that curves to the right of its intended path, and a hook curves off to the left. Suppose a golfer hits the ball from the seventh tee at the US Women’s Open and the shot is a 160 yard slice 4o from the path straight to the cup. If the tee is 177 yards from the cup, how far does the ball lie from the cup?

Example 4: Using the formula: Area = ½ bc sin A, find the area of the triangle if d = 4, e = 7, and c = 9.

Section 6.1 (mini-lesson) Example 1: Convert 315o to radian measure.   Change radians to degree measure.