Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- 18 trig substitutions by math266 434 views
- Excel annuity-lab by math260 609 views
- t1 angles and trigonometric functions by math260 881 views
- 4.5 calculation with log and exp by math260 2958 views
- 4.6 more on log and exponential equ... by math260 951 views
- 3.1 methods of division by math260 1132 views

2,362 views

Published on

No Downloads

Total views

2,362

On SlideShare

0

From Embeds

0

Number of Embeds

7

Shares

0

Downloads

0

Comments

0

Likes

7

No embeds

No notes for slide

- 1. Analytic Trigonometry<br />
- 2. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />
- 3. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />
- 4. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />
- 5. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />
- 6. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />
- 7. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />(cos(),sin())<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />
- 8. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />By Pythagorean Theorem,<br />y2 + x2 = 1, so we have:<br />sin2() + cos2()=1<br />or s2 + c2 = 1<br />or all angle .<br />(cos(),sin())<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />
- 9. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . <br />
- 10. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />
- 11. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />The Trig-Hexagram<br />
- 12. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />The Trig-Hexagram<br />the CO-side<br />the regular-side<br />
- 13. Analytic Trigonometry<br />The Division Relations:<br />
- 14. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />
- 15. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />II<br />III<br />I<br />
- 16. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />III<br />I<br />II<br />
- 17. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />I<br />II<br />III<br />
- 18. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br /> The Reciprocal Relations<br />
- 19. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />
- 20. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />III<br />II<br />I<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />Example B: sec(A) = 1/cos(A), <br />
- 21. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />I<br />II<br />III<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A) <br />
- 22. Analytic Trigonometry<br /> Square-Sum Relations<br />
- 23. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />
- 24. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />
- 25. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />
- 26. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />
- 27. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />The identities from this hexagram are called the<br />Fundamental Identities. <br />
- 28. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />The identities from this hexagram are called the<br />Fundamental Identities. <br />Weassume these identities from here on and list the most important ones below.<br />
- 29. Fundamental Identities<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />
- 30. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />
- 31. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />
- 32. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.<br />
- 33. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />we have the following co-relations.<br />
- 34. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />we have the following co-relations.<br />sin(A) = cos(90 – A)<br />sin(A) = cos(B)<br />
- 35. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />w have the following co-relations.<br />sin(A) = cos(90 – A)<br />tan(A) = cot(90 – A)<br />sin(A) = cos(B)<br />tan(A) = cot(B)<br />
- 36. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />w have the following co-relations.<br />sin(A) = cos(90 – A)<br />tan(A) = cot(90 – A)<br />sec(A) = csc(90 – A)<br />sin(A) = cos(B)<br />tan(A) = cot(B)<br />sec(A) = csc(B)<br />
- 37. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />
- 38. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />
- 39. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />Therefore cosine is an even function and sine is an odd function. <br />
- 40. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />Therefore cosine is an even function and sine is an odd function. <br />Since tangent and cotangent are quotients of sine <br />and cosine, they are also odd functions.<br />
- 41. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />
- 42. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> Frank Ma<br />2006<br />
- 43. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> Frank Ma<br />2006<br />
- 44. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> Frank Ma<br />2006<br />
- 45. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> Frank Ma<br />2006<br />
- 46. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1<br /> Frank Ma<br />2006<br />
- 47. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br /> Frank Ma<br />2006<br />
- 48. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> Frank Ma<br />2006<br />
- 49. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> = 1 – sin2(A) <br /> Frank Ma<br />2006<br />
- 50. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> = 1 – sin2(A) = cos2(A)<br /> Frank Ma<br />2006<br />
- 51. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />
- 52. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) <br />
- 53. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 <br />
- 54. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 ;square-relation<br />= tan2(x) <br />
- 55. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 ;square-relation<br />= tan2(x) ; division relation <br />= <br />sin2(x)<br />cos2(x)<br />

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment