Ch05+Trig+Functions

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=Chapter Five - The Trigonometric Functions= lists this chapter's objectives, topics, homework assignments, and assessments.

reviews briefly the three basic trigonometric ratios, SOHCAHTOA, and a few other points about trigonometry.

5-1 Angles and Their Measure

 * An angle may be generated by the rotation of two rays that share a fixed endpoint. An angle with its vertex at the origin and its initial side along the positive //x//-axis is said to be in **standard position**.
 * One ray may remain fixed to form the //initial side// of the angle.
 * The second ray may rotate to form the //terminal side//of the angle.
 * If the rotation is in a counterclockwise direction, the angle formed is a positive angle.
 * If the direction is clockwise, it is a negative angle.


 * If the terminal side of an angle in standard position coincides with one of the axes, the angle is called a **quadrantal angle**.
 * Angles measured in degrees can be expressed in decimal form (//e.g.//, 55.360 o ) or in degrees-minutes-seconds form (//e.g.//, 55 o 21' 36").
 * One degree is equal in measure to 60 minutes; one minute is equal in measure to 60 seconds.
 * Convert an angle ddd o mm' ss" into decimal form by performing the following calculation:
 * { ddd o mm' ss" } = { ddd + mm/60 + ss/3600 } o


 * Angles can also be measured in //**radians**//.
 * The radian measure of an angle in standard position is defined as the length of the corresponding arc on a unit circle.
 * An angle of 90 degrees is one quarter turn of a full circle. The full unit circle has an arc length of 2*pi, and a quarter of that is pi / 2, so an angle of 90 degrees is equivalent to ( pi / 2 ) radians.


 * **Coterminal angles** are those whose terminal rays are coincident.
 * If an angle has measure A degrees, then all angles of the form A + 360*k degrees (where k is an integer) are coterminal with A.
 * If an angle has measure B radians, then all angles of the form B+2pi*k radians (where k is an integer) are coterminal with B.

5-2 Central Angles and Arcs

 * The length of any circular arc, s, is equal to the product of the measure of the radius of the circle, r, and the radian measure of the central angle, //theta//:
 * s = r * //theta.//


 * If an object moves along a circle of radius r units, then its instantaneous linear speed, v, is given by r * (//theta// / t), where (//theta/// t) represents the angular velocity in radians per unit of time:
 * v = r * ( //theta// / t).


 * If //**theta**// is the measure of the central angle expressed in radians and **//r//** is the measure of the radius of the circle, then the area of the sector, //**A**//, is given by
 * //A// = 1/2 * //r// 2 * //theta.//
 * Consider what this formula becomes when //theta// is 2 pi radians.

5-3 Circular Functions

 * Watch [|this video] for an overview of basic trigonometric ratios.
 * Here is a short video that gives some tips for remembering key aspects of the unit circle: Tips For Memorizing The Unit Circle

5-4 Trigonometric Functions of Special Angles

 * Trigonometric ratios of the quadrantal angles (0, 90, 180, 270, and 360 degrees) are summarized in a table on page 264.
 * Some functions are undefined for certain **quadrantal** angles.
 * Quadrantal is a fancy word for describing the angles whose terminal side lie on either the //x//-axis or the //y//-axis.
 * Use your calculator to find the tangent of 90 degrees. How about the tangent of (3*pi / 2). What happens? Why?


 * A 45-45-90 triangle has sides with lengths in the ratio 1:1:sqrt(2).
 * A 30-60-90 triangle has sides with lengths in the ratio 1:sqrt(3):2.
 * From these ratios one can determine the trigonometric functions for several non-quandrantal angles whose measures are multiples of 30 or 45 degrees.
 * A table on page 265 summarizes the sine and cosine values for selected angles from 0 to 180 degrees.
 * Students would be //very wise// to familiarize themselves with the trigonometric ratios of these selected angles.

5-5 Right Triangles

 * [[file:Right Triangle Word Problems.pdf]] using Angle of Elevation and Angle of Depression.

5-6 The Law of Sines

 * [|Here] is a video that presents a proof of the Law of Sines.
 * [[file:Law of Sines textbook problems.pdf]]
 * [[file:Law of Sines word problems.pdf]]

5-7 The Law of Cosines

 * [|Here] is a video that presents a proof of the Law of Cosines.
 * Basically, the Law of Cosines is the Pythgorean Theorem with an "adjustment term" that goes into effect if the triangle is not a right triangle:
 * c 2 = a 2 + b 2 - 2ab * cos C
 * There are two equally valid variants of this formula. Here's one: a 2 = b 2 + c 2 - 2bc * cos A.
 * Note that when Angle C is a right angle, the formula collapses into the familiar c 2 = a 2 + b 2.
 * If Angle C is acute, however, cos C will be a positive number, causing the length c to be less than it would for a right triangle.
 * On the other hand, if Angle C is obtuse, the cos C will be negative, and subtracting it causes length c to increase.
 * [[file:Law of Cosines textbook problems.pdf]]

5-8 Area of Triangles
>> >>
 * If you know the lengths of two sides of a triangle (//b// and //c//), and the measure of the **included angle**, //A//, you can express the area, //K//, of the triangle as:
 * //K// = 1/2 * //bc// * sin //A//
 * Analogous formulas can be generated for different combinations of known sides and angles:
 * //K// = 1/2 * //ac// * sin //B//
 * K = 1/2 * a//b// * sin //C//
 * **Hero's Formula:** If the measures of the sides of a triangle are //a//, //b//, and //c//, then the area of the triangle, //K//, can be expressed as
 * //K// = sqrt[//s// (//s// - //a//)(//s// - //b//) (//s// - //c//)] where //s// = 1/2 * (//a// + //b// + //c//)
 * ======**Area of a Circular Segment:** If //alpha// is the measure of the central angle expressed in **radians** and the radius of the circle has a measure of r units, then the area of the segment, S, can be expressed as======
 * S = 1/2 * r 2 * (//alpha// - sin //alpha//).
 * [[file:Area textbook problems.pdf]]

Here are solutions to selected review problems from Sections 5-6 through 5-8:



>>
 * **The Lot Project**
 * The lot project will be the culmination of our Chapter 5 work. Each student will be given a plot plan of a building lot and, using our new knowledge of trigonometry, will calculate the area of the lot.
 * [[file:Lot Project Rubric.pdf]]
 * [[file:Lot Project plot plans .pdf]]
 * [[file:Lot 64 solution.pdf]]


 * Chapter Review**
 * [[file:Ch05 Review.pdf]]
 * [[file:Ch05 Review problem solutions.pdf]]

**Additional Chapter Review - Sections 5.6, 5.7, and 5.8**

 * Suggested textbook problems:
 * Section 5.6, #20, 22, 24, 26, 30
 * Section 5.7, #22, 24, 26, 28
 * Section 5.8, #24,26, 30, 34
 * Solution to above problems: [[file:Review 5-6 to 5-8 solutions.pdf]]


 * Chapter 5 Review**
 * Practice Test (19 questions)[[file:Ch05 Practice Test.pdf]]
 * Practice Test solutions: [[file:Ch05 Prac Test soln.pdf]]