Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rule of marteloio wikipedia , lookup
Perceived visual angle wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Euclidean geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
P.o.D. – Find the amplitude, period, and vertical translation of each graph. 1.) 𝑦 = 3 sin(2𝑥 ) − 5 2.) 𝑦 = cos(𝑥) 3.) 𝑦 = −4 sin(. 5𝑥 ) + 2 4.) 𝑦 = −2 cos(3𝑥 ) − 6 1 1 3 3 5.) 𝑦 = sin ( 𝑥) + 6.) 𝑦 = − cos(6𝑥 ) − 7 7.) 𝑦 = sin(𝑥 ) + 𝜋 10-7: The Law of Sines Learning Target(s): I can determine the measure of an angle given its sine, cosine, or tangent; find the missing sides and angles of a triangle using the Law of Sines; identify and use theorems relating sines and cosines; solve real-world problems using the Law of Sines. Essential Question: How do you use trigonometry to solve and find the areas of oblique triangles? Oblique Triangle = a triangle that has no right angle; could be acute or obtuse. Law of Sines: C a A c B EX: For triangle ABC, A=30 degrees, B=45 degrees, and a=32 feet. Find the remaining sides and angle. Always begin by drawing a picture. B 45 32 30 A C The easiest thing to find is the missing angle. Now, use the given information to find side b. Next, find side c. Finally, write your solution with all 6 parts of the triangle together. EX: Because of prevailing winds, a tree grew so that it was leaning 6 degrees from the vertical. At a point 30 meters from the tree, the angle of elevation to the top of the tree is 22.5 degrees. Find the height of the tree. h 96 22.5 30 Begin by finding the missing angle. Set up the Law of Sine to find h. EX: For the triangle ABC, a=12inches, b=5inches, and A=31 degrees. Find the remaining sides and angles. We need to find angle B. Next, find angle C by subtraction. Now, find side c using the Law of Sines. Write your final answer. The Ambiguous Case of the Law of Sines (SSA): Why couldn’t you prove triangle congruency with SSA in geometry? (Draw a picture on the whiteboard) Three possibilities will exist: Case I: If A is an acute angle: If 𝑎 < 𝑏 sin 𝐴, then ___ triangle exists. If 𝑎 = 𝑏 sin 𝐴, then exactly ____ right triangle exists. If 𝑎 > 𝑏 sin 𝐴, then ___ triangles exist. If 𝑎 ≥ 𝑏, then ____ triangle exists. Case II: If A is an obtuse angle: If a < b, then ___ triangle exists. If a > b, then ____ triangle exists. EX: Show that there is no triangle for which A=60degrees, a=4, and b=14. We are in Case I since A is acute. In Case I, if a<b sin A, then EX: Find two triangles for which A=58degrees, a=4.5, and b=5. Again, we are in Case I since A is acute. We now need to find b sin A. Since a>b sin A {4.5>4.24}, we know that we have ____ solutions. Begin by solving using the Law of Sines. Triangle #1: Use subtraction to find angle C. Now, use the Law of Sines to find side c. Therefore, the first triangle consists of: Triangle #2: In other words, the second angle B is the ______________ of the first angle B. Find angle 𝐶2 by subtraction. Lastly, find side 𝑐2 using the Law of Sines. Triangle #2 consists of: Area of a Triangle: The area of any triangle is onehalf the product of the lengths of two sides times the sine of their included angle. That is, EX: Find the area of a triangular lot containing the side lengths that measure 24 yards and 18 yards and form an angle of 80 degrees. *On your own, write a calculator program titled SASAREA that will compute the area of a triangle given two sides and the included angle. EX: On a small lake, a child swam from point A to point B at a bearing of 𝑁 28° 𝐸. The child then swam to point C at a bearing of 𝑁 58° 𝑊. Point C is 800 meters due north of point A. How many total meters did the child swim? (draw a picture and show the work on the whiteboard) EX: Suppose you are given a triangle with two angles of 42 degrees and 74 degrees and an included side of 18-m. Find the length of the side opposite the 74 degrees. (draw a triangle on the whiteboard and have a student come to the board to solve) EX: In triangle TRI, angle R is 25 degrees, angle I is 72 degrees, and side r is 13. Find side t. (draw a triangle on the board and have a student come to the board to solve) HW Pg.703 2, 7-12, 15-18 Worksheet 10.7B