Chapter 6 ADDITIONAL TOPICS IN TRIGONOMETRY

6.1 Law of Sines Objectives –Use the Law of Sines to solve oblique triangles –Use the Law of Sines to solve, is possible, the triangle or triangles in the ambiguous case –Find the area of an oblique triangle using the sine function –Solve applied problems using the Law of Sines

Law of Sines Previously, our relationships between sides of a triangle and the angles were unique only to RIGHT triangles What about other triangles? Obtuse or acute ones? (oblique – not right!) The following relationship exists (A,B,C are measures of the 3 angles; a,b,c are the lengths of sides opposite those angles):

Solving an oblique triangle If given: A = 50 degrees, B = 30 degrees, b = 7 cm. Can you solve this triangle? If so, is the solution unique? You know C = 100 degrees (Now that you know the measure of all 3 angles and the length of 1 side, how many triangles exist? Only one!) You can find “a” & “c” by law of sines:

What if given: a=5”,b=7”,B=45 degrees Law of Sines indicates sinA=.505. There are 2 angles,A, such that sin(A) =.505. A = 30 degrees, OR A = 150 degrees (WHY? Look at your unit circle!) Are there 2 possible triangles? NO – in this case, a (5”)is smaller than b (7”), and if angle A = 150 degrees, it must be opposite the longest side of the triangle. Clearly, it is not, therefore only 1 triangle exists. (continued )

Example continued If A = 30 degrees & B = 45 degrees, C=105 degrees Use law of sines again to find c.

What if given: a=7”,b=5”,B=45 degrees Law of Sines indicates sinA=.99. There are 2 angles,A, such that sin(A) =.99. A = 82 degrees, OR A = 98 degrees (WHY? Look at your unit circle!) Are there 2 possible triangles? YES – in this case, a (7”)is larger than b (5”), and if angle A = 82 or 98 degrees, it is a larger angle than B. Clearly, there are 2 triangles that exist. (continued)

2 possible triangles Triangle 1: use law of sines to find c: Triangle 2: use law of sines to find c:

Finding area of an oblique triangle Using Law of Sines it can be found that for any triangle, height (h) = b sin A (if c is considered to be the base), therefore Area= ½ c b sinA

6.2 Law of Cosines Objectives –Use the Law of Cosines to solve oblique triangles –Solve applied problems using the Law of Cosines –Use Heron’s formula to find the area of a triangle.

What if know a=4”,b=6”,C=70 degrees? Law of Sines does NOT apply Law of Cosines was developed: Use this to solve for c in the given triangle: b=c, so C = B = 70 degrees, thus A = 40 degrees

Using Law of Cosines, you can solve a triangle with 3 given sides If the 3 sides are given, only 1 such triangle exists

Heron’s Formula for Area of a Triangle If the 3 sides of a triangle are known, the area can be found (based on Law of Cosines):

6.3 Polar Coordinates Objectives –Plot points in the polar coordinate system –Find multiple sets of polar coordinate for a given point –Convert a point from polar to rectangular coordinates –Convert a point from rectangular to polar coordinates –Convert an equation from rectangular to polar coordinates –Convert an equation from polar to rectangular coordinates

Defining points in the polar system Location of a point is based on radius (distance from the origin) and theta (the angle the radius moves from standard position (positive x-axis in a cartesian system)) Any point can be described many ways. i.e. 2 units out moving pi/2 is the same as a radius of 2 units moving -3pi/2 or a radius of 2 units moving 5pi/2

What is the relationship between cartesian coordinates & polar ones? The radius = r is the hypotenuse of a rt. triangle that has base = x & height=y Thus, If x = horizontal leg & y = vertical leg of a right triangle, then

Find rectangular coordinates for

Convert a rectangular equation to a polar equation

6.4 Graphs of Polar Equations Objectives –Use point plotting to graph polar equations –Use symmetry to graph polar equations

Graphing by Point Plotting Given a function, in polar coordinates, you can find corresponding values for “r” and “theta” that will make your equation true. Plotting several points and connecting the points with a curve provides the a graph of the function. Example next page

Example: Put values in for theta that range from 0 to 2pi (once around the circle..after that the values begin repeating)

Continued example

“Special” curves generated by general forms

r=3cos(theta)

r=2+3sin(theta)

r=3cos(5theta)

6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem Objectives –Plot complex numbers in the complex plane –Fine absolute value of a complex # –Write complex # in polar form –Convert a complex # from polar to rectangular form –Find products & quotients of complex numbers in polar form –Find powers of complex # in polar form –Find roots of complex # in polar form

Complex number = z = a + bi a is a real number bi is an imaginary number Together, the sum, a+bi is a COMPLEX # Complex plane has a real axis (horizontal) and an imaginary axis (vertical) 2 – 5i is found in the 4 th quadrant of the complex plane (horiz = 2, vert = -5) Absolute value of 2 – 5i refers to the distance this pt. is from the origin (continued)

Find the absolute value Since the horizontal component = 2 and vertical = -5, we can consider the distance to that point as the same as the length of the hypotenuse of a right triangle with those respective legs

Expressing complex numbers in polar form z = a + bi

Express z = i in complex form

Product & Quotient of complex numbers

Multiplying complex numbers together leads to raising a complex number to a given power If r is multiplied by itself n times, it creates If the angle, theta, is added to itself n times, it creates the new angle, (n times theta) THUS,

Taking a root (DeMoivre’s Theorem) Taking the nth root can be considered as raising to the (1/n)th power Now finding the nth root of a complex # can be expressed easily in polar form HOWEVER, there are n nth roots for any complex number & they are spaced evenly around the circle. Once you find the 1 st root, to find the others, add 2pi/n to theta until you complete the circle

If you’re working with degrees add 360/n to the angle measure to complete the circle. Example: Find the 6 th roots of z= i Express in polar form, find the 1 st root, then add 60 degrees successively to find the other 5 roots.

6.6 Vectors Objectives –Use magnitude & direction to show vectors are equal –Visualize scalar multiplication, vector addition, & vector subtraction as geometric vectors –Represent vectors in the rectangular coordinate system –Perform operations with vectors in terms of i & j –Find the unit vector in the direction of v. –Write a vector in terms of its magnitude & direction –Solve applied problems involving vectors

Vectors have length and direction Vectors can be represented on the rectangular coordinate system Vectors have a horizontal & a vertical component A vector starting at the origin and extending left 2 units and down 3 units is given below, along with the magnitude of the vector (distance from the origin):

Adding & subtracting vectors Scalar Multiplication Add (subtract) horizontal components together & add (subtract) vertical components together v = 3i + 7j, w = -i + 2j, v – w = 4i + 5j Scalar Multiplication: multiply the i & j components by the constant v = 3i + 7j, 4v = 12i + 28j (the new vector is 4 times as long as the original vector)

Unit Vector has the same direction as a given vector, but is 1 unit long Unit vector = (original vector)/length of vector Simply involves scalar multiplication once the length of the vector is determined (recall the length = length of hypotenuse if legs have lengths = a & b) Given vector, v = -2i + 7j, find the unit vector:

Writing a Vector in terms of its Magnitude & Direction v is a nonzero vector. The vector makes an angle measured from the positive x-axis to v, and we can talk about the magnitude & direction angle of this vector:

Velocity Vector: vector representing speed & direction of object in motion Example: The wind is blowing 30 miles per hour in the direction N20 degrees E. Express its velocity as a vector v. If the wind is N20 degrees E, it’s 70 degrees from the positive x-axis, so the angle=70 degrees and the magnitude is 30 mph.

Resultant Force Adding 2 force vector together Add horizontal components together, add vertical components together

6.7 Dot Product Objectives –Find dot product of 2 vectors –Find angle between 2 vectors –Use dot product to determine if 2 vectors are orthogonal –Find projection of a vector onto another vector –Express a vector as the sum of 2 orthogonal vectors –Compute work.

Definition of Dot Product The dot product of 2 vectors is the sum of the products of their horizontal components and their vertical components

Find the dot product of v&w if v=3i+j and w= -2i - j

Properties of Dot Product If u,v, & w are vectors and c is scalar, then

Angle between vectors, v and w

Parallel Vectors Parallel: the angle between the vectors is either 0 (the vectors on top of each other) or 180 (vectors are in opposite directions), in either case, cos(0)=1, cos(180) = -1, this will be true if the dot product of v & w = (plus/minus)product of their magnitudes

Orthogonal Vectors Vectors that are perpendicular to each other. The angle between vectors is 90 degrees or 270 degrees. cos(90)=cos(270)=0 Since they are orthogonal if the numerator = 0, thus the dot products of the 2 vectors = 0 if they are orthogonal

Vector projection of v onto w

Work done by a force F moving an object from A to B Force and distance are both vectors