Exam #4 Review Section 7.1 What you should know:  Be aware that we are now solving oblique triangles (not a right triangle).  You should know how to label a triangle properly (side a is opposite angle A, side b is opposite angle B, side c is opposite angle C).  Know the three angles of a triangle sum to 180° .  Know that when working with triangles, we measure the angles in degrees.  The largest angle of a triangle is opposite the longest side. The smallest angle of a triangle is opposite the shortest side. The middle‐valued angle is opposite the middle‐valued side.  Memorize the Law of Sines. You can use either form in my class (I do not care). or 



You should know that the Law of Sines works for the SAA or ASA case really well. The SSA case is discussed in 7.2 and the Law of Sines is used, but it may lead to more than one triangle in the solution. You might need to review how to find bearing. Know how to find bearing when it is measured clockwise, from the north. Remember in this form you are only given the degree measure of one angle. Know how to find bearing when it starts with North or South, gives an angle, and then gives East or West. Example: The bearing is N 36 W . 
You should know how to find the area of a triangle using
, and that to find the height we often need to use the sine function. Section 7.2 What you should know:  This section is also about the applications of the Law of Sines, but this time the SSA case is studied. This means two sides and the non‐included angle.  The SSA case may result in 0 triangles, 1 triangle, or 2 possible triangles. You need to state how many triangles are formed, and find all values for all possible triangles formed. Please see the yellow box at the top of page 316 for the conditions.  In the SSA case, first solve for the angle between the two given sides. Then subtract this angle from 180° and see if the 3 angles still sum to180° . If so, two possible triangles are formed. Section 7.3 What you should know:  Memorize the Law of Cosines: 


2
2
2
Know that the Law of Cosines is used for the SAS and SSS cases. Know that there is no “AAA” postulate! Do not memorize Heron’s Formula for the area of a triangle, but do know how to use it. See page 323. Section 7.4 What you should know:  A vector is a quantity with both a magnitude and a direction. Velocity and force are typical examples of vectors.  Know that a vector is drawn as a directed line segment. The length of the line segment represents the magnitude. The direction is usually represented by an angle drawn from the horizontal.  To write the vector between point O and point P, we can use boldface (when typing) OP, or draw a horizontal line above the letters (when writing by hand) .  Vectors are also commonly named v, u, and w.  The magnitude of a vector is represented with vertical bars, like an absolute value:| |.  A scalar is a quantity with only a magnitude, no direction. Scalars are represented by real numbers. Speed is an example of a scalar: 45 mph does not tell you what direction the object is traveling in.  You should know how to add two vectors, find the opposite of a vector, and perform scalar multiplication on a vector.  If a vector v is placed so the its initial point is at the origin, and its terminal point is (a, b), 〈 , 〉. Here a is called the horizontal component of the vector, and b then we can write is called the vertical component of the vector. Such vectors are often called position vectors. 
When 


0. Make sure you can do examples like 1, 2 and 3 in Section 7.4. You should be able to do application problems like those I assigned in section 7.4. Know how to perform vector operations on position vectors. See the box at the top of page 338. 〈 , 〉, then | |
√
and the direction angle satisfies , where 





〈1,0〉 and 〈0,1〉. Using this, we can rewrite any Know how to use the notation 〈 , 〉
vector . Notice the “i” here is in bold and denotes a vector, not the imaginary number i. A unit vector has length 1. 〈 , 〉∙〈 , 〉
. The dot product between two vectors is: ∙
The dot product between two vectors is an operation where you input two vectors and it outputs a scalar. 180° , between any The dot product can be used to compute the angle , where 0°
two vectors. ∙
cos
| || |
You do not need to memorize the angle formula just above, but you do need to know how to use it. Review Problems for Chapter 7 You should be able to do problems 1‐47 in the Chapter 7 Review Exercises on pages 355‐357. Section 8.1 What you should know: 



The imaginary unit is defined as: √ 1. A complex number in standard form is: , where a is the real part and b is the imaginary part. Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. You should know how to find complex solutions to quadratic equations. See examples 2 and 3 on page 363. You should know how to add, subtract, multiply and divide complex numbers. To add and subtract: distribute the sign if needed and combine like terms. 
To multiply: first rewrite any radicals of the form √


write √
√ . Then multiply using the distributive property or FOIL as indicated. Here, assume that 0. The complex conjugate of is . To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Then simplify both the numerator and denominator. To find powers of , you should know: 


as a complex number. So, first √ 1 1 1 After the 4 power, the pattern repeats. See page 368 for details. th
Section 8.2 What you should know:  Be able to graph a complex number 
Know that if , then in the complex plane. , and tan
,
0. So the trig form is °


. Here we choose 0
360° . Be careful to select the angle to be in the correct quadrant. First graph the value of . Be able to write a complex number in standard form in trig (or polar) form. If a complex number is given in trig form, you can convert it to rectangular form. If , then and , and is in rectangular form. Section 8.4 What you should know:  Know how to use De Moivre’s Theorem to find powers of complex numbers. I will give you this theorem (see below), do not memorize it. .  If , then Review Problems for Chapter 8 Do problems 1‐20, and 25‐39 on pages 418‐419 in the Chapter 8 Review.