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Trigonometry
- 1. A mathematics PowerPoint by Eric Zhao
- 2. Trigonometry is the study and solution of Triangles. Solving a triangle means finding the value of each of its sides and angles. The following terminology and tactics will be important in the solving of triangles. Pythagorean Theorem (a2 +b2 =c2 ). Only for right angle triangles Sine (sin), Cosecant (csc or sin-1 ) Cosine (cos), Secant (sec or cos-1 ) Tangent (tan), Cotangent (cot or tan-1 ) Right/Oblique triangle
- 4. A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent. Let us look at this triangle a c b 唏 A B C Given the assigned letters to the sides and angles, we can determine the following trigonometric functions. The Cosecant is the inversion of the sine, the secant is the inversion of the cosine, the cotangent is the inversion of the tangent. With this, we can find the sine of the value of angle A by dividing side a by side c. In order to find the angle itself, we must take the sine of the angle and invert it (in other words, find the cosecant of the sine of the angle). Sin慮= Cos 慮= Tan 慮= Side Opposite Side Adjacent Side Adjacent Side Opposite Hypothenuse Hypothenuse = = = a b c a b c
- 5. Try finding the angles of the following triangle from the side lengths using the trigonometric ratios from the previous slide. 6 10 8 慮 A B C 留 硫 Click for the Answer The first step is to use the trigonometric functions on angle A. Sin 慮 =6/10 Sin 慮 =0.6 Csc0.6~36.9 Angle A~36.9 Because all angles add up to 180, B=90-11.537=53.1 C 2 34尊 A B 留 硫 The measurements have changed. Find side BA and side AC Sin34=2/BA 0.559=2/BA 0.559BA=2 BA=2/0.559 BA~3.578 The Pythagorean theorem when used in this triangle states that BC2 +AC2 =AB2 AC2 =AB2 -BC2 AC2 =12.802-4=8.802 AC=8.8020.5 ~3
- 7. When solving oblique triangles, simply using trigonometric functions is not enough. You need The Law of Sines C c B b A a sinsinsin == The Law of Cosines a2 =b2 +c2 -2bc cosA b2 =a2 +c2 -2ac cosB c2 =a2 +b2 -2ab cosC It is useful to memorize these laws. They can be used to solve any triangle if enough measurements are given. a c b A B C
- 8. When solving a triangle, you must remember to choose the correct law to solve it with. Whenever possible, the law of sines should be used. Remember that at least one angle measurement must be given in order to use the law of sines. The law of cosines in much more difficult and time consuming method than the law of sines and is harder to memorize. This law, however, is the only way to solve a triangle in which all sides but no angles are given. Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.
- 9. a=4 c=6 b A B C 28尊 Solve this triangle Click for answers Because this triangle has an angle given, we can use the law of sines to solve it. a/sin A = b/sin B = c/sin C and subsitute: 4/sin28尊 = b/sin B = 6/C. Because we know nothing about b/sin B, lets start with 4/sin28尊 and use it to solve 6/sin C. Cross-multiply those ratios: 4*sin C = 6*sin 28, divide 4: sin C = (6*sin28)/4. 6*sin28=2.817. Divide that by four: 0.704. This means that sin C=0.704. Find the Csc of 0.704 尊. Csc0.704尊 =44.749. Angle C is about 44.749尊. Angle B is about 180-44.749-28=17.251. The last side is b. a/sinA = b/sinB, 4/sin28尊 = b/sin17.251尊, 4*sin17.251=sin28*b, (4*sin17.251)/sin28=b. b~2.53.
- 10. a=2.4 c=5.2 b=3.5A B C Solve this triangle: Hint: use the law of cosines Start with the law of cosines because there are no angles given. a2 =b2 +c2 -2bc cosA. Substitute values. 2.42 =3.52 +5.22 -2(3.5)(5.2) cosA, 5.76-12.25-27.04=-2(3.5)(5.2) cos A, 33.53=36.4cosA, 33.53/36.4=cos A, 0.921=cos A, A=67.07. Now for B. b2 =a2 +c2 -2ac cosB, (3.5)2 =(2.4)2 +(5.2)2 -2(2.4)(5.2) cosB, 12.25=5.76+27.04-24.96 cos B. 12.25=5.76+27.04-24.96 cos B, 12.25-5.76-27.04=-24.96 cos B. 20.54/24.96=cos B. 0.823=cos B. B=34.61. C=180-34.61-67.07=78.32.
- 12. Trigonometric identities are ratios and relationships between certain trigonometric functions. In the following few slides, you will learn about different trigonometric identities that take place in each trigonometric function.
- 13. What is the sine of 60尊? 0.866. What is the cosine of 30尊? 0.866. If you look at the name of cosine, you can actually see that it is the cofunction of the sine (co-sine). The cotangent is the cofunction of the tangent (co-tangent), and the cosecant is the cofunction of the secant (co-secant). Sine60尊=Cosine30尊 Secant60尊=Cosecant30尊 tangent30尊=cotangent60尊
- 14. Sin 慮=1/csc 慮 Cos 慮=1/sec 慮 Tan 慮=1/cot 慮 Csc 慮=1/sin 慮 Sec 慮=1/cos 慮 Tan 慮=1/cot 慮 The following trigonometric identities are useful to remember. (sin 慮)2 + (cos 慮)2 =1 1+(tan 慮)2 =(sec 慮)2 1+(cot 慮)2 =(csc 慮)2
- 16. Degrees and pi radians are two methods of showing trigonometric info. To convert between them, use the following equation. 2 radians = 360 degrees 1 radians= 180 degrees Convert 500 degrees into radians. 2 radians = 360 degrees, 1 degree = 1 radians/180, 500 degrees = radians/180 * 500 500 degrees = 25 radians/9
- 18. Write out the each of the trigonometric functions (sin, cos, and tan) of the following degrees to the hundredth place. (In degrees mode). Note: you do not have to do all of them 1. 45尊 2. 38尊 3. 22尊 4. 18尊 5. 95尊 6. 63尊 7. 90尊 8. 152尊 9. 112尊 10. 58尊 11. 345尊 12. 221尊 13. 47尊 14. 442尊 15. 123尊 16. 53尊 17. 41尊 18. 22尊 19. 75尊 20. 34尊 21. 53尊 22. 92尊 23. 153尊 24. 1000尊
- 19. Solve the following right triangles with the dimensions given 5 c 22 A B C 9 20 18 A B C A a c 13 B C 52 尊 c 12 8 尊 A B C
- 20. Solve the following oblique triangles with the dimensions given 12 22 14A B C a 25 b 28 尊 A B C 31 尊 15 c 24 35 尊 A B C 5 c 8A B C 168 尊
- 21. 1. 45尊 2. 38尊 3. 22尊 4. 18尊 5. 95尊 6. 63尊 7. 90尊 8. 152尊 9. 112尊 10. 58尊 11. 345尊 12. 221尊 13. 47尊 14. 442尊 15. 123尊 16. 53尊 17. 41尊 18. 22尊 19. 75尊 20. 34尊 21. 53尊 22. 92尊 23. 153尊 24. 1000尊 Find each sine, cosecant, secant, and cotangent using different trigonometric identities to the hundredth place (dont just use a few identities, try all of them.).
- 23. Convert to degrees 3.2 rad 3.1 rad 1.3 rad 7.4 rad 6.7 rad 7.9 rad 5.4 rad 9.6 rad 3.14 rad 6.48 rad 8.23 rad 5.25 rad 72.45 rad 93.16 rad 25.73 rad 79.23 rad 52.652 rad 435.96 rad 14.995 rad 745.153 rad
- 24. Creator Eric Zhao Director Eric Zhao Producer Eric Zhao Author Eric ZhaoMathPower Nine, chapter 6Basic Mathematics Second edition By Haym Kruglak, John T. Moore, Ramon Mata-Toledo