1. C HAPTER 13 Further Work with Trigonometry

2. S ECTION 13-1 T HE S ECANT, C OSECANT, AND C OTANGENT F UNCTIONS The three reciprocal trigonometric functions are: secant (sec), cosecant (csc), and cotangent (cot) Secant is the reciprocal of cosine Cosecant is the reciprocal of sine Cotangent is the reciprocal of tangent There are restrictions on the reciprocal trigonometric functions because you cannot divide by zero

3. In a right triangle, you get the following ratios: Open your book to page 807. We are going to look at the green box at the bottom. Remember that when you graph, you have to watch your mode and your window! Ex1. A ladder is placed at a 60° angle with the ground. To the nearest tenth of a foot, how long must the ladder be to reach 27 feet up the side of the building? Show your set up using an reciprocal trigonometric ratio.

4. Ex2. Find exactly Open your book to page 808. We are going to look at the graphs of secant and cotangent.

5. S ECTION 13-2 P ROVING T RIGONOMETRIC I DENTITIES One way to visually demonstrate that an identity is true is to graph both sides of the identity and the graphs should coincide See Pythagorean Identity graphs on pg. 811 You can use known identities to derive new ones See example 1 on page 812 One way to prove that an equation is an identity Start with one side of a proposed identity Rewrite it using definitions, known identities, or algebraic properties until one side equals the other There are often multiple ways to prove identities to be true Every question is different, so take your time and plot your course

6. Another proof technique is to take each side and rewrite them in different ways until you get two equal expressions If you use this technique, don’t use an equal sign until then end because you are not sure the two are equal Draw a vertical line instead See top of page 813 A third (and final) proof technique is to begin with a known identity and derive statements equivalent to it until the proposed identity appears Carefully read all of the examples Ex1. Prove that for all x,

7. S ECTION 13-4 P OLAR C OORDINATES Where rectangular coordinates are given by (x, y), polar coordinates are written [r, θ] The length of the segment is r The angle of rotation is θ Start from the same position as with the unit circle As with the trigonometric coordinates, there are infinitely many ways to write the polar coordinates of a point You can use the positive or negative length You can add multiples of π or 2π to the angles

8. Graph each point [r, θ] Ex1. Ex2. Ex3.

9. Open your book to page 822. We are going to look at example 3. You must use exact values whenever possible To convert from polar to rectangular: The x-value is found by r ·cosθ The y-value is found by r · sinθ Ex4. Find the rectangular coordinates for the point [3, 150°] in exact form

10. To convert from rectangular coordinates to polar coordinates Find the radius by: Either the positive or the negative possibility can work Find theta by: Consider which quadrant the original point lies in and use that to determine which r and which θ to use (see page 824) Ex5. Find a set of polar coordinates for the point (6, -1)

11. S ECTION 13-5 P OLAR G RAPHS To use your graphing calculator: Make sure your mode is POLAR (not FUNC) Enter your equation into y = (it now looks like r = and uses θ in stead of x) Adjust your window accordingly Your θ values should be between 0 and 2π (or 0° and 360°) Your θ-step value should be about.13 (or 7.5) Your x values should be between -4 and 4 (or so) Your y values should be between -4 and 4 (or so) Open your book to page 827. We are going to look at example 1.

12. Ex1. Graph r = cos 2θ on your graphing calculator Open your book to page 829. We are going to look at the graph near the top of the page. Graphs like these are called petal curves because they look like flower petals You can graph the coordinates of polar graphs on a rectangular graph as long as you go in order of the θ values The rectangular graphs are the curves we started seeing in chapter 4

13. Ex2. Graph all [r, θ] for r = 2cos 3θ

14. S ECTION 13-6 T HE G EOMETRY OF C OMPLEX N UMBERS You can graph complex numbers on a rectangular coordinate plane The horizontal axis is the real axis and the vertical axis is the imaginary axis (and they should be labeled as such) The graph will be a single point (real, imaginary) When you plot two complex numbers, their sum, and the origin on a complex plane and connect them, it forms a parallelogram (see page 833)

15. Ex1. Graph each of the following complex numbers in the complex plane A) 3 + 6i B) -4 – 5i C) 7i

16. To change complex numbers to polar form: Find the radius by: Find θ by: Watch the quadrant again! For any complex number z = [r, θ] with r > 0, r is its distance from the origin (called the absolute value or the modulus of the complex number) It is found by: Ex2. Find the polar coordinates for -7 + 7 i