Use Trigonometry with Right Triangles SECTION 13.1

Transcription

1 Use Trigonometry with Right Triangles SECTION 13.1

2 WARM UP In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values (leave in radical form). a = 6, b = 8 c = 10, b = 7 If you walk 2.0 km due east and then 1.5 km due north, how far will you be from your starting point? Answer 2.5 km Write the expression in simplest form. 180

3 WHY STUDY TRIGONOMETRY? Hipparchus of Nicaea ( BC) Father of Trigonometry Studied Astronomy more than 2000 years ago Hippocrates of Chois ( BC) and Erathosthenes of Cyrene ( BC) paved the way using triangle ratios that were used by Egyptian and Babylonian engineers 4000 years earlier. The term trigonometry emerged in the 16th century from greek roots: Tri = three gonon = side metros = measure

4 WHY STUDY TRIGONOMETRY? Trig functions arose from the consideration of ratios within right triangles. The ultimate tool for engineers in the ancient world. As knowledge progressed from a flat earth to a world of circles and spheres, trig became the secret to understanding circular phenomena. Circular motion let to harmonic motion and waves. electrical current modern telecommunications stored sound wave digitally on a CD

5 WHY STUDY TRIGONOMETRY? Advent of Calculus made Trig functions more important than ever Every kind of periodic (recurring) behavior can be modeled by simply combining sine functions Modeling aspect of trig functions is another focus of study

6 WHAT YOU LL LEARN ABOUT Use trigonometric functions to find lengths Evaluate trigonometric functions Use special angles to find lengths What a radian is How to convert from degrees to radians and from radians to degrees How to find the arc length of a circle How to find the area of a sector of a circle

7 SOH CAH TOA Let θ be an acute angle of a right triangle, then sin θ = opposite cos θ = adjacent tan θ = Opposite hypotenuse hypotenuse adjacent csc θ = hypotenuse sec θ = hypotenuse cot θ = adjacent opposite adjacent opposite Notice that ratios in second row are reciprocals of the ratios in the first row.

8 SOH CAH TOA Let θ be an acute angle of a right triangle, then csc θ = sec θ = cot θ =

9 EVALUATE THE SIX TRIGONOMETRIC FUNCTIONS OF THE ANGLE Θ 7 θ 24

10 IF Θ IS AN ACUTE ANGLE OF A RIGHT TRIANGLE AND COS Θ = 3/8, FIND THE VALUE OF THE OTHER FIVE FUNCTIONS.

11 FIND THE VALUE OF X IN THE RIGHT TRIANGLE SHOWN. 45 x

12 FIND THE VALUE OF X IN THE RIGHT TRIANGLE SHOWN. x

13 SOLVE TRIANGLE ABC Make sure your calculator is set to degree mode. A 54 b C c = 20 a B

14 From a point on the ground 28 feet from the base of a flagpole, the angle of elevation to the top of the flagpole is 63 degrees. Estimate the height of the flagpole. Angle of elevation is the angle formed by the line of sight to the object and a line parallel to the ground

15 RADIAN A central angle of a circle has measure of 1 radian if it intercepts an arc with the same length as the radius. Because the circumference of a circle is 2πr, there are 2π radians in a circle. Thus 360⁰ = 2π and 180⁰ = π

16 DEGREE/RADIAN MEASURES

17 AN ANGLE IS IN STANDARD POSITION WHEN THE VERTEX IS AT THE ORIGIN AND THE INITIAL SIDE IS ON THE POSITIVE X-AXIS. Draw an angle in standard position with following measurements π -5π/2

18 Coterminal angles are angles with the same initial and terminal sides, but different measurements. Find one positive and one negative angle that are coterminal with: π/ π/3

19 DEGREE-RADIAN CONVERSION To convert radians to degrees, multiply by 180 π radians To convert degrees to radians, multiply by π radians 180

20 DEGREE-RADIAN CONVERSION Convert 75⁰ to radians Convert -5π/4 radians to degrees

21 ARC LENGTH AND AREA OF A SECTOR Arc length: s = rθ (θ measured in radians) Area: 1 2 A= r θ 2 (θ measured in radians)

22 Children at a day camp are playing a game on a circular field. The shaded sector in the figure is called the safe zone, and is marked off by rope along its outer edge. Find the length of the rope and the area of the safe zone if the radius of the circle is 18 ft and the angle of the sector is 120⁰.

23 HOMEWORK Pg. 856: 3 28 every other pair, 30-32, 36 Pg. 862: 3-5, 7 44 every other pair, 49, 52