8.1 Non-Right Triangels

Law of Sines

  1. Use this picture to prove the Law of Sines.

  1. In the triangle shown here, solve for the unknown sides and angle

  1. Find the elevation of the UFO from the beginning of the section.

  1. In the triangle shown here, solve for the unknown sides and angles.

When using the law of signs to solve for angles, we must be careful! Solving for an angle may give 2 answers, one answer, or no answers!

  1. Find all possible triangles if one side has length 4 opposite an angle of 20° and a second side has length 8.
  2. Find all possible triangles if one side has length 4 opposite an angle of 20° and a second side has length 2.
  3. Find all possible triangles if one side has length 4 opposite an angle of 20° and a second side has length 10.
  4. Make up your own angle, side, side problem that has 2 solutions.
  5. Make up your own angle, side, side problem that has 1 solution.
  6. Make up your own angle, side, side problem that has no solutions.

Law of Cosines

  1. Use this picture to prove the Law of Cosines.

The Law of Cosines is useful for solving for a missing side length when you know adjacent “side/angle/side” (SAS) values.

  1. Find the unknown side and angles of this triangle.

  1. Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles. How far from port is the boat?

The Law of Cosines is also useful (and needed!) to solve when you know all three side lengths. Notice in this case, you have only one possible angle for your solution!

  1. Solve for the angle \(\theta\) in the triangle shown.

  1. Given \(\alpha=25\), \(b=10\), and \(c=20\), find the missing side and angles.

  2. On many cell phones with GPS, an approximate location can be given before the GPS signal is received. This is done by a process called triangulation, which works by using the distance from two known points. Suppose there are two cell phone towers within range of you, located 6000 feet apart along a straight highway that runs east to west, and you know you are north of the highway. Based on the signal delay, it can be determined you are 5050 feet from the first tower, and 2420 feet from the second. Determine your position north and east of the first tower, and determine how far you are from the highway.

  1. To measure the height of a hill, a woman measures the angle of elevation to the top of the hill to be 24 degrees. She then moves back 200 feet and measures the angle of elevation to be 22 degrees. Find the height of the hill.

8.2 Polar Coordinates

  1. Plot \((r,\theta)=(3,\frac{\pi}{6})\).
  2. Plot \((r,\theta)=(4,\frac{4\pi}{3})\).
  3. Plot \((r,\theta)=(-2,\frac{\pi}{4})\).

  1. Find the Cartesian coordinates for the point \((r,\theta)=(5,\frac{2\pi}{3})\).
  2. Find the polar coordinates of point with Cartesian coordinates \((-3,-4)\).
  3. Sketch the graph of \(r=\theta\).
  4. Sketch the graph of \(r=4\cos(\theta)\).
  5. Sketch the graph of \(r=4\sin(\theta)+2\).
  6. Sketch the graph of \(r=\cos(3\theta)\).
  7. Rewrite the Cartesian equation \(x^2+y^2=6y\) as a polar equation.
  8. Rewrite the polar equation \(r=2\sin(\theta)\) in Cartesian form.

Exam 3

8.3 Polar Form of Complex Numbers

The most basic complex number is \(i\), defined to be \[i = \sqrt{−1}\], commonly called an imaginary number.

Any real multiple of i is also an imaginary number.

A complex number is a number \[z = a +bi\], where a and b are real numbers

a is the real part of the complex number
b is the imaginary part of the complex number

To plot a complex number like \(3− 4i\), we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane.

  1. Plot the number \(3− 4i\) on the complex plane.
  2. Add \(3− 4i\) and \(2 + 5i\).
  3. Subtract \(2 + 5i\) from \(3− 4i\).
  4. Multiply: \((2 − 3i)(1 + 4i)\).
  5. Divide: \(\frac{2+5i}{4-i}\).

  1. Find the polar form of -8.
  2. Find the polar form of \(-\sqrt{2}+\sqrt{2}i\).

Below is an important formula relating the exponential and trigonometric functions.

  1. Prove \(e^{\pi i}=-1\).
  2. Write \(3e^{\frac{\pi}{6}i}\) in \(a+bi\) form.
  3. Evaluate \((-\sqrt{2}+\sqrt{2}i)^8\).

8.4 Vectors

A vector is an object that has both a length and a direction. (Vector means “carrier” in Latin)

Geometrically, a vector can be represented by an arrow that has a fixed length and indicates a direction.

Algebraically, vectors can be represented in two ways:

  • If, starting at the point A, a vector moves toward point B, we write \(\overrightarrow{AB}\) to represent the vector.
  • A vector may also be indicated using a single letter in boldface type, like u, or by capping the letter representing the vector with an arrow, like \(\vec{u}\).
  1. Draw a vector that represents the movement from the point P(-1, 2) to the point Q(3,3)

To geometrically scale a vector by a constant, scale the length of the vector by the constant. Scaling a vector by a negative constant will reverse the direction of the vector.

  1. Draw any vector call it \(\vec{u}\). Based on this, draw \(2\vec{u}\), \(-\vec{u}\), and \(\frac{1}{2}\vec{u}\).

  2. What is \(3\vec{u}-4\vec{u}\)?

  3. Draw two vectors, \(\vec{u}\) and \(\vec{v}\). Then draw \(\vec{u} + \vec{v}\) and \(\vec{u} - \vec{v}\).

A vector \(\vec{u}\) can be described by its magnitude, or length, \(|\vec{u}|\), and an angle \(\theta\).

A vector with length 1 is called a unit vector.

  1. Find the component form of a vector with length 7 at an angle of 135 degrees.

  1. Write your answer in #5 using alternate notation.

  2. Find the magnitude and angle \(\theta\) of the vector \(\vec{u}=\langle 3, -2 \rangle\).

  3. An object is launched with initial velocity 200 meters per second at an angle of 30 degrees. Find the initial horizontal and vertical velocities.

  1. Given \(\vec{u}=\langle 3,-2 \rangle\) and \(\vec{u}=\langle -1,4 \rangle\), find:
  1. \(\vec{w} = \vec{u}+\vec{v}\)
  2. \(\vec{w} = 3\vec{u}-2\vec{v}\)

  1. A woman leaves home, walks 3 miles north, then 2 miles southeast. How far is she from home, and what direction would she need to walk to return home? How far has she walked by the time she gets home?

  2. In a scavenger hunt, directions are given to find a buried treasure. From a starting point at a flag pole you must walk 30 feet east, turn 30 degrees to the north and travel 50 feet, and then turn due south and travel 75 feet. Sketch a picture of these vectors, find their components, and calculate how far and in what direction you must travel to go directly to the treasure from the flag pole without following the map.

8.5 Dot Product

  1. Use the component definition of the dot product to prove the geometric version.

  2. Find \(\langle 3,-2 \rangle \cdot \langle 3,-2 \rangle\).

  3. An object is being pulled up a ramp in the direction \(\langle 5,1 \rangle\) by a rope pulling in the direction \(\langle 4,2 \rangle\). What is the angle between the rope and the ramp?

  1. Calculate the angle between the vectors \(\langle 6,4 \rangle\) and \(\langle −2,3 \rangle\).

Note: If the dot product of two vectors is zero, that means they are perpendicular (orthogonal), and vice versa!

  1. Are the two vectors orthogonal? \(\langle -7,3 \rangle\) and \(\langle −2,6 \rangle\).

  1. A cart is pulled 20 feet by applying a force of 30 pounds on a rope held at a 30 degree angle. How much work is done?

  2. Find the work down moving an object from the point (1, 5) to (9, 14) by the force vector \(\vec{F} = \langle 3,2 \rangle\)