Week 10 of m151
(§ 2, Mr. Lane)
3–7 April 2017

Math Learning Center (in Math 011) has tutors:
9am–4pm on Monday–Thursday and 9am–1pm on Friday

math@Mansfield (Library) has tutors: 6:30pm – 9:00pm on Sunday–Thursday

math@Mansfield (Library) has tutors: 6:30pm – 9:00pm on Sunday–Thursday

- Monday [d30] (3 April 2017)
WeBWorK:
RQ-8-2a closes at 10 am
- Study section 8.1 (
*Trig Functions and Right Triangles*).- Some of this is an alternative presentation of stuff in the Trigonometry chapter of the U MT edition for our text.
- Notice where inverse-trig functions (arcsin, arccos, arctan) are used to compute angles in a right triangle.

- Work on these exercises and problems in section 8.1:
1, 7, 11, 19, 21;
27, 35, 39,
**41** - Study section 8.2 (
*Non-Right Triangles*) through Example 2**and**the important sentence at top of page 337.- Figure 8.26 (page 335) uses a
traditional labeling for sides and angles:
- side
*a*is opposite angle*A*(or angle α) - side
*b*is opposite angle*B*(or angle β) - side
*c*is opposite angle*C*(or angle γ)

**Law of Cosines**depends on that systematic labeling. - side
- Consider Law of Cosines applied to a right triangle.
Remember that cosine of a right angle is zero.
This
**special case for Law of Cosines**has a familiar name:*Pythagorean Theorem*.

- Figure 8.26 (page 335) uses a
traditional labeling for sides and angles:
- Work on these exercises and problems in section 8.2: 2, 6, 9, 11, 13; 44, 36

- Study section 8.1 (
- Tuesday [d31]
WeBWorK:
RQ-8-2b closes at 10 am,
day-29 closes at 11 pm
- Study the remainder of section 8.2:
*Law of Sines*(begins on page 337).- The triangle in Figure 8.29 (page 337) has
- height
**h = b · sin(C)** - area
**(1/2) · a · h = (1/2) · a · b · sin(C)**

**(1/2) · b · c · sin(A)**and**(1/2) · c · a · sin(B)** - height
- After reading about
*The Ambiguous Case*(page 338), pick two side-lengths and an angle for which there will be two possible triangles; then find the third side-length for each triangle. - Example 2 discusses a situation where an unambiguous angle can be found using the Law of Cosines.

- The triangle in Figure 8.29 (page 337) has
- Work on more exercises and problems in section 8.2:
1, 7, 17, 21, 27, 13;
**32**, 33, 43,**45** - Anticipate a quiz on Tuesday or Wednesday.

- Study the remainder of section 8.2:
- Wednesday [d32]
WeBWorK:
RQ-8-3a closes at 10 am,
day-30 closes Thursday at 11 pm
- Study section 8.3 (
*Polar Coordinates*) through Example 5.- Each polar coordinate specifies a unique point.
The point with polar coordinates
**(r,θ) = (a,b)**has rectangular coordinates_{polar}**x =****r · cos(θ)****= a · cos(b)****y =****r · sin(θ)****= a · sin(b)** - Each point has many polar coordinates.
- The origin has polar coordinates (0,theta) for any value of theta.
- A
**negative radial**component is best interpreted as the corresponding positive distance measured in the**opposite**direction.- (4, 1), (−4, 1 + π), (4, 1 + 2 π), (−4, 1 − 3 π) are polar coordinates for the same point.
- This is consistent with the formulas to compute rectangular coordinates as functions of polar coordinates because cos(t + π) = −cos(t) and sin(t + π) = −sin(t) for all values of t.
- Several graphs on page 346 show the authors are wrong
when they write
"
*but we shall not do so*" on page 344 (at the end of the last sentence before Example 3).

- Each polar coordinate specifies a unique point.
The point with polar coordinates
- Polar Coordinate Graph Paper is available
(as a downloadable PDF) from many sites, for example,
- incompetech.com/graphpaper/polar/
- www.waterproofpaper.com/graph-paper/polar-graph-paper.shtml

- Work on these exercises and problems in section 8.3:
1, 5, 9, 11, 15,
**17**; 23,**24**, 33, 37 & 38

- Study section 8.3 (
- Friday [d33]
WeBWorK:
RQ-8-3b closes at 10 am,
day-31 closes Saturday at 11 pm
- Write-up solutions of these four problems —
show your work and interpret your results. Hand-in your written
homework at the
**start**of Friday's class.**section**8.1 8.2 8.3 **problems**36 34, 42 40 - Resume study of Polar Coordinates by
reviewing Examples 4 & 5 of §8.3,
then study the remainder of that section.
- Analysis of the polar graph for
**r = f(θ)**may be aided by also examining the rectangular graph of**y = f(x)**. - As you study Examples 6 and 7, identify which part(s) of each
curve corresponds to positive values of
**r**and which corresponds to negative values of**r**. - After studying the text's discussion of Example 6,
work on modified versions with
**sin**and**cos**swapped, e.g., r = 3 cos(2 θ) and r = 4 sin(3 θ) - Most graphing calculators can plot a curve given by some
polar coordinate equations.
- The polar equation must have the form r = f(θ)
- Use the Window button to specify interval for θ (in addition to horizontal and vertical dimensions of the viewing rectangle).

- Analysis of the polar graph for
- Work on more exercises and problems in section 8.3:
12, 15, 19,
**21**; 39 & 41, 42 & 43, 46, 47, 49

- Write-up solutions of these four problems —
show your work and interpret your results. Hand-in your written
homework at the

© Richard B Lane
Last modified: 31 March 2017, Friday 06:37