5.1 Circles

Clicker Question:

  1. Does a perfect circle exists in real life? Why or why not?

    A: Yes
    B: No way!

  2. How do you find the distance between two points on a grid? For example, find the distance between the points \((-3,2)\) and \((2,5)\)?

  3. Can you prove the Pythagorean Theorem?

  1. Can you prove the distance formula?

  1. Use the distance formula to find the distance between \((1,6)\) and \((3,-5)\).

  2. Use the distance formula to find the distance between \((3,-5)\) and \((1,6)\).

Mathematically speaking, a circle is all the points that are a same set distance \(r\) (called the radius) from one point \(c\) (called the center).

If \(c=(0,0)\) of a circle with radius \(r\), any point \((x,y)\) on this circle satisfies the equation: \[r = \sqrt{(x-0)^2+(y-0)^2}\] Or more simply:

\[r^2 = x^2+y^2.\]

  1. What is the radius of the circle graphed below?

Write an equation for this circle.

Now if \((h,k)\) is the center of a circle, then \(r = \sqrt{(x-h)^2+(y-k)^2}\), or: \[(x-h)^2+(y-k)^2=r^2.\]

Write an equation for a circle centered at the point \((1, -2)\) with radius 5.

  1. Write an equation for a circle centered at the point \((-3, 2)\) with radius 4.

    A: \((x-3)^2+(y+2)^2 = 4\)
    B: \((x-3)^2+(y+2)^2 = 16\)
    C: \((x+3)^2+(y-2)^2 = 4\)
    D: \((x+3)^2+(y-2)^2 = 16\)

  2. Find the x intercepts of the circle above.

  3. Find the equation of circle at the point \((2, 4)\) that passes through \((6,6)\)

  4. Determine if \(x^2 + y^2 = 16\) and \(y = 3 - x\) intersect. If so, find their intersection.

  5. An ant is heading toward a circular region of pancake batter. The ant cannot breathe while walking through the batter. The batter has a radius of 4 cm. The ant is going to walk from a point 6 cm due north of the batter’s center to a point 5 cm due west of the batter’s center. What distance will the ant have to walk through the batter

  6. If the ant moves at 0.8 cm/sec, how long will it have to hold its breath?

5.2 Angles

  1. Does pineapple belong on pizza?

    A: Yes
    B: Never C: Depends on the occasion

We commonly use Greek letters to represent angles:

  • \(\alpha = alpha\)
  • \(\beta = beta\)
  • \(\gamma = gamma\)
  • \(\theta = theta\)
  • \(\phi = phi\)

This is great because it allows us to avoid confusion between linear measurements (\(a,b,c,...,x,y,z\)) and angular measurements (\(\alpha, \beta,\gamma,...\))

Notice that not all angles are not unique: \(0^{\circ}=360^{\circ}\).

  1. Find an angle \(\alpha\) that is coterminal with \(-45^{\circ}\) , where \(0^{\circ} \leq \alpha < 360^{\circ}\).

Clicker Question:

  1. Find an angle \(\alpha\) that is coterminal with \(870^{\circ}\) , where \(0^{\circ} \leq \alpha < 360^{\circ}\).

Clicker Question:

  1. Find an angle \(\alpha\) that is coterminal with \(-870^{\circ}\) , where \(0^{\circ} \leq \alpha < 360^{\circ}\).

The circumference or length around a circle is \(2\pi r\).

  1. What is the circumference around a unit circle, (a circle of radius 1)?

  2. What is the arc length of:

    1. \(180^\circ\) of a unit circle?
    2. \(90^\circ\) of a unit circle?
    3. \(60^\circ\) of a unit circle?
    4. \(30^\circ\) of a unit circle

  3. Answer the (a)-(d) above for a circle of radius 3.

  1. Find the radian measure of
    1. \(180^\circ\)
    2. \(90^\circ\)
    3. \(60^\circ\)
    4. \(30^\circ\)

If we know the radian measure of any angle, we can use that to find arc length!

  1. Find the arclength along a circle of radius 10 subtended by an angle of 215 degrees.

Clicker Question:

  1. Mercury orbits the sun at a distance of approximately 36 million miles. In one Earth day, it completes 0.0114 rotation around the sun. If the orbit was perfectly circular, what distance through space would Mercury travel in one Earth day?

Note the relationship between radians and degrees:

  1. Convert \(\frac{\pi}{6}\) radians to degrees.

Clicker Question:

  1. Convert 15 degrees to radians.

Below are the common angles in degrees. In calculus, we more frequently consider angles in radians.

  1. Convert/label each of these angles in radians.

  1. On a circle of radius 6 feet, what angle in degrees would subtend an arc of length 3 feet? *

The area of a circle is \(A=\pi r^2\). What is the area of a sector subtended by an angle of: a. \(\pi\) (half of a circle) b. \(\pi/2\) (quarter of a circle) c. \(\pi/4\) (quarter of a circle) d. \(\theta\) (any sector!)

Clicker Question:

  1. An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees. What is the area of the sector of grass the sprinkler waters?

  1. A water wheel completes 1 rotation every 5 seconds. Find the angular velocity in radians per second.

  1. Your bicycle has wheels 24-in.-diameter wheels and your tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed your are traveling down the road.

Clicker Question:

  1. You upgraded you bicycle to 26-in.-diameter wheels and just biked a distance of 1.5 miles in 6 minutes. Find the average angular velocity of your wheels in rad/min. How many revolutions per minute do your wheels make? *

  2. You are standing on the equator of the Earth (radius 3960 miles). What is your linear and angular speed (relative to the center of the earth)?

5.3 Points on Circles

A distress signal is sent from a sailboat during a storm, but the transmission is unclear and the rescue boat sitting at the marina cannot determine the sailboat’s location. Using high powered radar, they determine the distress signal is coming from a distance of 20 miles at an angle of 135 degrees from the marina. How many miles east/west and north/south of the rescue boat is the stranded sailboat?

This situation can be modeled below, where \(r=20\), \(\theta=135^\circ\), and we need to find \(x\) and \(y\).

The answer to this question lies in some patterns we can observe in triangles. Consider the two triangles below, inscribed in circles of different radius, but having the same angle \(\theta\).

  1. Prove both triangles above have the same corresponding angles.

Triangles that have the same angles are called similar. In this case, the ratio of corresponding side lengths are the same.

For example, \(\frac{x_1}{y_1}=\frac{x_2}{y_2}\).

  1. List as many different was as you can to combine \(x\), \(y\), and \(r\) to make ratios.

Any ratio we listed will be the same for two similar triangles, regardless of their size!

For right triangles, each of these ratios actually has a name. Here are the two most popular.

  1. The point \((3, 4)\) is on the circle of radius \(5\) with some angle \(\theta\). Find \(\cos(\theta)\) and \(\sin(\theta)\).
  2. Find \(\cos(90^\circ)\) and \(\sin(90^\circ)\).
  3. Find cosine and sine of the angle \(\pi\).

We can also rewrite the definition above to find \(x\) and \(y\) in terms of \(r\) and \(\theta\).

  1. Plug in \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\) into \(x^2+y^2=r^2\) to prove the Pythagorean identity.

  1. If \(\sin(\theta)=\frac{3}{7}\) and \(\theta\) is in the second quadrant, find \(\cos(\theta)\). (This is the same thing as asking, if \(P=(x,y)\) is a point on the unit circle and \(y=\frac{3}{7}\), what is \(x\)?)
  2. If \(\sin(\theta)=\frac{2}{8}\) and \(\theta\) is in the fourth quadrant, find \(\cos(\theta)\).

Next, how can we actually find values for \(\sin(\theta)\) and \(\cos(\theta)\) given an angle \(\theta\)??

First, consider \(\theta=45^\circ=\frac{\pi}{4}\). Note in this case, \(cos(\theta)=sin(\theta)\).

  1. Use the Pythagorean identity to find \(\sin(\frac{\pi}{4})\) and \(\cos(\frac{\pi}{4})\).
  2. Find the coordinates of the point on a circle of radius 6 and angle \(\frac{\pi}{4}\).

Next, let’s consider \(\theta=30^\circ=\frac{\pi}{6}\). Notice how we can copy the triangle made by the \(30^\circ\) angle and make a equilateral triangle:

  1. Use this to find \(\sin(30)\) and \(\cos(30)\).
  2. Use symmetry to find \(\sin(60)\) and \(\cos(60)\). (See picture below)

Now we’ve found sine and cosine for \(\theta = 30,45,60,90^\circ\). We can use symmetry to find other angles too!

  1. Find the values of all the points \((x,y)=(\cos(\theta),\sin(\theta))\) on your unit circle (see problem 2.1 #12)

To make the process of finding angles, we call the angle the helps apply our work in the first quadrant that angle’s reference angle:

  1. Find the reference angle of 150 degrees. Use it to find cos( 150°) and sin( 150 ° ) .

  2. Use this idea to fill out the rest of your unit circle. Try to do it on your own, then check your work by comparing your answers the unit circle on page 370 of the textbook.

  3. Find the coordinates of the point on a circle of radius 12 at an angle of \(\frac{7\pi}{6}\).

  4. Find the coordinates of the point on a circle of radius 5 at an angle of \(\frac{5\pi}{3}\).

  5. Answer the sailboat question the beginning of the chapter! (I.e. What are the coordinates (relative to the marina), of a boat 20 miles away at an angle \(135^\circ\) from the marina?)

Note: When using your calculator to find \(\sin\) and \(\cos\), make sure you always check whether you are in radians or degree mode! If you forget how, this video will show you how to check and switch between radians and degrees, or vice versa.

5.4 Other Trig Functions

All the other ratios of triangle side lengths we listed in problem 2 in the previous section have names too!

  1. Use these definitions of the trigonometric functions and the definitions of sine and cosine (\(x=r \cos(\theta)\), and \(y=r \sin(\theta)\)) to prove the following identities:
    1. \(\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\)
    2. \(\sec(\theta)=\frac{1}{\cos(\theta)}\)
    3. \(\csc(\theta)=\frac{1}{\sin(\theta)}\)
    4. \(\cot(\theta)=\frac{1}{\tan(\theta)}=\frac{\cos(\theta)}{\sin(\theta)}\)
  2. Evaluate:
    1. \(\tan(45^\circ)\)
    2. \(\cot(45^\circ)\)
    3. \(\sec(\frac{5 \pi}{6})\)
    4. \(\csc(\frac{5 \pi}{6})\)
  3. If \(\sin(\theta)=\frac{2}{7}\) and \(\theta\) is in quadrant II, find:
    1. \(\cos(\theta)\)
    2. \(\sec(\theta)\)
    3. \(\csc(\theta)\)
    4. \(\tan(\theta)\)
    5. \(\cot(\theta)\)
  4. Use a calculator to find sine, cosine, and tangent of the following values:
    1. \(0.5\)
    2. \(15.2\)
    3. \(10^\circ\)
    4. \(105^\circ\)
  5. Use the definition of the trig functions and their identities to simplify the following expressions:
    1. \(\frac{\sec(t)}{\tan(t)}\)
    2. \(\csc(t)\tan(t)\)
    3. \(\frac{1+\sin(t)}{1+\csc(t)}\)
    4. \(\frac{\sin^2(t)+\cos^2(t)}{\cos(t)}\)

Sometimes these are useful in proving trig identities. Note that in proving an identity, do not treat the identity like an equation to solve and try to multiple or divide the same things to both sides of the equation. Start from one side and try to manipulate it until you arrive at the other side.

  1. Prove the following identities:
    1. \(\frac{1+\cot(\theta)}{\csc(\theta)} = \sin(t) + \cos(\theta)\)
    2. \(\frac{cos^2(\theta)}{1+sin(\theta)} = 1 - \sin(\theta)\)
    3. \(\frac{\sec(\theta)-\cos(\theta)}{\sin(\theta)} = \tan(\theta)\)

Hint: on \(b\) and \(c\), use the Pythagorean Identity (\(\cos^2(\theta)+\sin^2(\theta)=1\)) to substitute \(\cos^2(\theta)\) for something in terms of sine.

  1. Use the Pythagorean Identity to prove:
    1. \(1+\tan^2(\theta)=\sec^2(\theta)\)
    2. \(1+\cot^2(\theta)=\csc^3(\theta)\)

These variants of the Pythagorean Identity are often very useful.

  1. Use these variants of the Pythagorean Identity to prove the following:
    1. \(\frac{\tan(\theta)}{\sec^2(\theta)-1}=\cot(\theta)\)
    2. \(\frac{1}{\sec(\theta)-1}-\frac{1}{\sec(\theta)+1}=2\cot(\theta)\)

5.5 Right Triangle Trigonometry

Recall the definition of sine and cosine.

With these definitions, we can actually separate the triangle from the circle. As long as we have a right triangle, the same idea applies:

SOH-CAH-TOA !!

  1. Given the triangle shown below, find the value for \(\cos(\alpha)\), \(\sin(\alpha)\), and \(\tan(\alpha)\).

  1. Given the triangle shown, find the value for \(\cos(\alpha)\), \(\sin(\alpha)\), \(\cos(\beta)\), and \(\sin(\beta)\).

  1. Find the unknown sides of the triangle pictured here.

  1. To find the height of a tree, a person walks to a point 30 feet from the base of the tree, and measures the angle from the ground to the top of the tree to be 57 degrees. Find the height of the tree.

  1. A person standing on the roof of a 100 foot tall building is looking towards a skyscraper a few blocks away, wondering how tall it is. She measures the angle of declination from the roof of the building to the base of the skyscraper to be 20 degrees and the angle of inclination to the top of the skyscraper to be 42 degrees.

  1. Draw a picture of an angle \(\theta\) in the first quadrant, and the angle \(\beta=\frac{\pi}{2}-\theta\). Conjecture relationships between: \(\sin(\theta)\), \(\cos(\theta)\), \(\cos(\beta)\), and \(\sin(\theta)\).
  1. Find the length x.

  1. Find the length x.

  1. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is \(80^\circ\). How high does the ladder reach up the side of the building?

  2. A 400 foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(18^\circ\) and that the angle of depression to the bottom of the tower is \(3^\circ\). How far is the person from the monument?