Presentation on theme: "Welcome to Trigonometry! Well be Getting Triggy with these concepts… 6.1: find exact values of trigonometric functions (5-1) 6.2: find coterminal and."— Presentation transcript:
Welcome to Trigonometry! Well be Getting Triggy with these concepts… 6.1: find exact values of trigonometric functions (5-1) 6.2: find coterminal and reference angles and to covert between units of angle measure (5-1) 6.2: 6.3: solve for missing values in right triangles (5-4, 5-5) 6.3: 6.4: use the law of sines and cosines and corresponding area formulas (5-6) 6.4: 6.5: use the ambiguous case of the law of sines to solve problems (5-7) 6.5:
6.2 find coterminal and reference angles and to convert between units of angle measure (5-1) In this section we will answer… How are angles measured in Trig? What are the different units of angle measure within degree measurement? What does it mean for angles to be co-terminal? How can I find a reference angle?
Angles and Their Measures From Geometry:
In Trig Angles are always placed on the coordinate plane. The vertex is at the origin and one side (the initial side) lies along the x-axis. The other side (the terminal side) lies in a quadrant or on another axii. This is called Standard Position.
Angle Direction: Angles can be measured in two directions. Counter-clockwise is positive. Clockwise is negative.
Degree Measurement: One full rotation = _________________. The circle has been cut into 360 equal pieces. Measure of less than a degree can be shown 2 ways: Decimal pieces: 55.75º Minutes and seconds: used for maps 103º 45 5 Each degree is divided into 60 minutes. Each minute is divided into 60 seconds. 1º = 60 = 3600
Change P280 #19 – 65 odd
Change 29º 30 60
Change 103º 12 42
Degrees on the Coordinate Plane: Unit Circle
Translating Rotations to Degrees Give the angle measure which is represented by each rotation: 5.5 rotations clockwise 3.3 rotations counterclockwise
Coterminal Angles Angles in standard position which share the same terminal side. 150º - 210º
Finding Coterminal Angles Simply add or subtract 360º as many times as you like. To write a statement to find EVERY angle coterminal with a certain angle:
Identify all the angles which are coterminal with the given angle. Then find one positive and one negative coterminal angle. 86º 294º
If each angle is in standard position, a) State the quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0º and 360º. 595º -777º
Reference Angle: The acute angle formed by the terminal side of an angle in standard position and the x-axis. The quickest route to the x-axis.
Recap: How are angles measured in Trig? What are the different units of angle measure within degree measurement? What does it mean for angles to be co- terminal? How can I find a reference angle?
6.3: solve for missing values in right triangles (5-4, 5-5) In these sections we will answer… How are the 6 trig ratios expressed in geometry? In trig? How can I use these relationships to solve triangle problems?
Right Triangles in Geometry: ABC Used the 3 basic trig ratios: sin A, cos A and tan A. SOH-CAH-TOA Now we will add 3 reciprocal ratios: csc A, sec A and cot A.
Solving Using Right Triangles:
*Be Careful with Reciprocal Ratios*
Try some more… p 288
Right Triangles in Trig: Angles are in Standard Position in the Unit Circle. 1 1
Try Some… The terminal side of angle θ in standard position contains (8,-15), find the 6 trig ratios. Now find the angle.
If the csc θ = -2 and θ lies in QIII, find all 6 trig values. Now find the angle.
If the tan θ = -2 and θ lies in QII, find all 6 trig values. Now find the angle.
Homework: P 288 #11 – 25 odd P 296 #15 – 45 odd and 49
6.1: find exact values of trigonometric functions (5-2/5-3) In this standard we will… Review the side relationships of 30°-60°-90° and 45°-45°-90° triangles. Build trig ratios based 30°-60°-90° and 45°- 45°-90° triangles.
Special Triangles from Geometry: Build a chart to show all 6 trig ratios.
Special Triangles on the Unit Circle:
6.3 solve for missing values in right triangles (5-4) In this standard we will answer… How can right triangle relationships by used to solve problems?
Lets start with some triangles… If A = 37º and b = 6, solve the rest of the triangle.
If B = 62º and c = 24, solve the triangle.
The apothem of a regular pentagon is 10.8 cm. Answer the following. Find the radius of the circumscribed circle. What is the length of one side of the pentagon? Find the perimeter of the pentagon. a = 10.8 cmr
p 303 Mr. Fleming is flying a kite. Ms Case notices the string makes a 70˚ angle with the ground. I know the string is 65 meters long, says Ms Case. I wonder how far is the kite above the ground? Ranger Gladd sights a fire from his fire tower in Alvarez National forest. He finds an angle of depression to the fire of 22˚. If the tower is 75 meters tall, how far is the fire from the base of the tower?
Partner Solve: ONE piece of paper. One person solves then second person checks and either praises or coaches. Change jobs. Do p 301 #1 – 9 all
Homework: LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE! P 303 #11-29 odd
6.3: solve for missing values in right triangles (5-5) In this section we will answer… What can I do to solve if I dont know any angles, just sides?
Solve each equation if
Inverse/arc trig ratios: To show you want to inverse or undo a trig ratio in order to get an angle there are two notations:
Evaluate each expression assuming a Quadrant I angle.
Architecture: Many cities place restrictions on the height and placement of skyscrapers in order to protect residents from completely shaded streets. If a 100-foot building casts an 88- foot shadow, what is the angle of elevation to the sun?
Partner Workout! P 309 #1 –14 all One piece of paper, take turns solving. If you arent solving you are the cheerleader/spotter. Encourage and save them from falling on their face.
Homework: LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE! P309 #15 – 45 odd
Section 5-6: The Law of Sines In this section we will answer… Is there some way to solve triangles that arent right triangles? How can I find the area of a triangle if I dont know its height?
The Law of Sines Up to now we have worked with RIGHT triangles, but what about other kinds?
The Law of Sines Lets look at ABC Then the following is true: A b c a C B
Using the Law of Sines How does it work? How many values do you have to be provided with? When wont it work?
From Geometry: AAS: A = 40º, B = 60º, and a = 20 SAS: b = 10, C = 50º, and a = 14 ASA: c = 2.8, A = 53º, and B = 61º One more for fun! b = 16, A = 42º, and c = 12
Area of a Triangle How did we find the area of triangles is geometry? You can now find the area of ANY triangle whether or not the height is given! A b c a C B h
Area of a Triangle Watch this! A b c a C B h
Area of a Triangle for SAS
What if I dont have 2 sides? What if I have 2 angles? Lets say I know b, C and A.
Finding the Area of a SSS Triangle: Can we do this? How?
For AAS or ASA:
Our Heros Formula! It saves us!
Try a couple… p 316 #20, 22, 24 and 26
Homework: p 316 #11 – 33 odd Mini-Quiz! on special triangle values everyday!!!
Section 5-7: The Ambiguous Case for the Law of Sines In this section we will answer… When can I use Law of Sines? Is there ever a case where ASS actually WORKS? How can I determine when I can use this really inappropriate acronym? Do I have to memorize the chart?
When can I use Law of Sines? If I have… AAS or ASA, always works! If I have… SAS or SSS, never works!
Is there ever a case where ASS actually WORKS? Exploration:
Our Nifty, Triangle Info Sheet! Do I have to memorize the chart?
Try a few… p 324 #11, 16, 19, 23 and 25
Homework: Mini-Quiz! on special triangle values tomorrow! Unit Test! on Tuesday