Not quite sure what this little kitten was going on about?
Well, the central angle is one way we can use to measure circles.
As the name suggests, it is an angle with the vertex at circle’s center. Like this:
Here, we can see that the vertex is the center of the circle, with points A and B on the outer edge. Each side of angle ∠AOB is a radius, which means that it is half of the diameter.
The outer edge of the circle between points A and B—the kitten’s pie crust—is an arc. The arc length is the measurement between A and B along the curve.
Okay, now that we know the terms cold, let’s set up some relationships.
We can make up some ratios. The arc length from AB is a portion of the entire circumference. There is a direct ratio between the circumference of a circle and its measurement of degrees. The A circle has 360°. The central arc is a portion of that.
Let’s say we know the arc length was 25π and the circumference was 166π. We can calculate the central angle easily!
What if we only knew the length of one side of an angle and the arc length?
This is a bit more challenging, but it’s just an extra step.
Let’s recap with our original angle & circle:
Say OA was 6 feet. Arc length is 1.4π
We can calculate the circumference of the circle: Circumference = Diameter x π
The radius is half the diameter, and we know that AO is a radius because it extends from the center of the circle at point O and reaches the outer edge at point A.
Circumference = Diameter x π = 2(radius) x π Circumference = 2(6 feet) x π Circumference = 12π
Great job! Now, we can plug in the numbers into our original formula!
Yep, pretty small angle, all right. Let’s hope that’s not the one our kitty ended up with! Stay tuned to keep learning on online tutoring platform StudyGate.
The half-angle formula, and its counterpart, the double-angle formula, will usually crop up in trigonometry, and sometimes pre-calculus classes. Sometimes you’ll be instructed to evaluate a trig function of an angle that’s not specifically included on the unit circle, and by using the identity of an angle that is included, you can still evaluate the function. For example, if you’re asked to find sin105º, an angle that’s not on the unit circle, you can use a half-angle identity and substitute 210º, which is on the unit circle, for 105º. The half-angle formulas for sine, cosine, and tangent are as written:
Follow this step-by-step process to use the half angle formula successfully.
If we’re trying to find sin105º, we first have to recognize that 105º is half of 210º, which is featured on the unit circle. We’d then rewrite the function as sin(210/2). Notice that the sine equation has the ± symbol, so we need to determine whether this function will be positive or negative. 105º lies in the second quadrant of the unit circle, so it will be positive. We then substitute x for the full angle value of 210. We then look to the unit circle again to find the cosine value of 210º, which is -3/2, and substitute that value for cos210. The final step is to simplify the function if possible. In this case, we can, because the denominators can be simplified.
While the formula seems tricky at first, it can be incredibly helpful in evaluating angle functions that are not listed on the unit circle. Always remember to be mindful of your plus and minus signs, as a function can transform into something completely bizarre and unintended otherwise. It also pays to check your simple addition, subtraction, and multiplication; mastering the formula isn’t worth much if you mix up your simple equation procedures. With the right amount of practice, the possibilities are limitless! StudyGate provides online learning resources to attain academic excellence.
Triangles come in all sizes. Some are acute, some are obtuse, and some are just right.
The ones that are just right weren’t designed by Goldilocks. But she knew a right triangle when she saw it. She knew that a triangle was “right” when it had a 90° angle—perfect for drawing sharp corners. Like these:
But there was an even more special right triangles. She could hardly wait to contain herself when she saw it. It was the awesome 45 45 90 triangle. Here it is in its mighty glory:
What’s so spectacular about this triangle? So many things it’s hard to list them all!
The two sides making up the 90° angle have the same length
The triangle is half of a square, as you can see on the right.
That means its hypotenuse (the diagonal of the square) is the side x √2
Pretty easy, huh?!
The tangent of 45-45-90 triangles is always 1.
Can’t get any easier than that!
The angles are easy to remember: 45+45+90 = 180°
The sides and hypotenuse are easy: the two sides are the same! x = x
While poets may sing the praise of angles, let’s set it down in basic terms.
What’s an angle?
An angle is the measurement of degrees between two lines.
These are all angles. They are acute angles because the gaps between the two lines are less than 90o. Acute means small, and small things are cute. Aren’t they cute?
These are angles, too. They are obtuse angles because the gaps between the two lines are GREATER than 90o. Obtuse means “slow” and “unintelligent.” While we don’t want to be mean to our angles, we can think of them as a bit unwieldly and hard to manage. Why are these angles so obtuse, darn it?!?
This means that angles that are EXACTLY 90o are called right angles. As Goldilocks would tell us, not too large, not too small, just right. We can make little squares inside them to show that the corner is 90o sharp. Pretty cool, huh?
OKAY, now that we have that downpat, what do we do with our happy angle family?
We can use angles to measure degrees.
For designers like architects, builders, and engineers, this is essential stuff…and for obvious reasons. If you’re measuring space and distance, whether for landscaping or for algorithms, angles are a must.
Let’s throw some basic calculations on how to measure angles.
These are pretty simple. It’s literally two angles next to each other! They share a common vertex (the point where they intercept) and one side. Here’s one:
For example, we see that the blue angle,∠BAD, has a measurement of 58o. The smaller angle ∠BAC has a measurement of 32o. What is the measurement of ∠CAD?
58o – 32o = 26o
A complementary angle is a specific type of adjacent angle. These angles add up to 90o to form a RIGHT ANGLE. Here’s one:
Since we know that the sum of the angles is always 90o, we can easily find out what one angle measurement is if we know the other. Since both angles are smaller than 90o, we know that all angles in complementary angles are acute.
Supplementary Angles take things a bit further. When we put two COMPLEMENTARY ANGLES together, they form a straight line, like this:
See how CO bisects DA? The angles on the right of ∠COA add up to 90o, right? That means that ∠DOC is also 90o. Put them together and we have supplementary angles. Angles that are supplementary share the same vertex and one line. They form a straight line and add up to 180o. Dead on arrival, no?
Here’s another example of a supplementary angle, this time, without complementary angles.
Note that if one angle is less than 90o, the other must be greater than 90o to add up to 180o. Thus, unless the angles are right angles, supplementary angles include one ACUTE and one OBTUSE angle.
We can also complicate this by stating that in a circle, all the angles add up to 360. This is because two right angles sharing a vertex forms a straight line, but travels half a circle. If we join two more right angles at the same vertex, the degrees of measurement would go all the way around! Like this:
Got it? Right on target!
Now let’s mess it up a bit with multiple sets of angles sharing one line, kinda like a road map. Line Y is the transversal line because it joins Lines A and B by crossing them:
We can see that [D] and [E] are supplementary since they share the same vertex, one line (Line Y) and add up to 180o. What else can we say?
[E] and [Q] are corresponding angles because they share the same exact position on their parallel lines A and B. Their angles are the same. The same with [F] and [R], [D] and [P], and [G] and [S]. Let’s highlight them:
See how the angle pairs correspond. What else can we say?
Alternate Interior Angles
We also have Alternate Interior Angles. Being interior angles, they are on the “inside,” between the parallel lines. These angles share the central transversal line, but have two different vertices. Think of them forming a letter Z and a reverse Z. In the example above, [G] and [Q] are alternate angles. So are [F] and [P]. Since Lines A and B are parallel,we can see that alternate angles have the same degrees.
Vertically Opposite Angles
From Z to X!They are angles that share the same vertex and transversal, but no corresponding side. They also have the same degrees. Think of them as the Letter X, with the angles opposite each other. In the above example, [E] and [G] are Vertically Opposite Angles. They each have 80o. So are [D] and [F]. Each one has 100o. Check ‘em out below:
Now that you’re seeing crosses, we’ll have a quick quiz. Find all the angles below:
We only have a single degree measurement, 111o, but this is no problem!
Solve the SUPPLEMENTARY ANGLE first!
A pair of supplementary angles add up to 180o, right? We also know that 111o is OBTUSE, so its adjacent angle, ∠C, must be ACUTE!
∠C + 111o = 180o
∠C = 69o
From there, we can fill out the top set of angles pretty easy.
∠A is a Vertically Opposite Angle from 111o. We know vertically opposite angles have the same degree measurement. So ∠A is 111o
The same with ∠C and ∠B.
∠C is 69o so ∠B is also 69o.
To check, we can also ADD ∠A + ∠B to see if they add up to 180o. (They do!)
We can simply copy and paste the top set of angles of the bottom, but we also use our vast knowledge of angle pairings to figure ‘em out. You’re an expert—go right ahead!
∠G is a corresponding angle with 111o.
That means ∠G =111o
∠D is an alternate interior angle with 111o. They form the inverse Z shape!
That means ∠D =111o
We can either figure out what ∠E and ∠F are since we know their supplementary angles, which add up to 180o
Or, we can look at them in pairs.
∠E is a corresponding angle with∠B. Since ∠B = 69o, ∠E is also 69o.
∠F is a Vertically Opposite Angle from∠E. Since we just figured out what ∠E is, and we know vertically opposite angles have the same value, ∠F is also 69o.
∠A = 111o.
∠B = 69o.
∠C = 69o.
∠D = 111o.
∠E = 69o.
∠F = 69o.
∠G = 111o.
Congrats! You are now the Awesome Angle Master! Have a glimpse of Reference Angle.
First, a minor let down. A trapezoid is nothing exotic as double summersault through the air. But it is pretty nifty. This is a trapezoid: What’s a trapezoid, then? It’s a shape with four sides. Of the four sides, two of them are parallel.
In this case, the top and bottom lines are parallel. The side lines will eventually intersect if they continue downward.
A trapezoid’s side lengths don’t matter. Two sides can be the same, as shown above. Or they can all be different, like these:
The top and bottom lines are still parallel, even though no two lines have the same length.
***Burning Question: Is a rectangle a trapezoid? Nope. A trapezoid can only have ONE set of parallel lines. Rectangles (and squares) have two sets! (Now ask yourself if a square can be a rhombus. To find out: see my next blog! 😊)
***Fun note: There is also a shape called a TRAPEZIUM, which is a four-sided shape with no parallel sides, like this:
This is not a trapezoid.
Now that we have a trapezoid, what do we do with it? How can find its area?
Why do we need to find its area? It’s like the old saying goes:
There was an old woman who lived in a shoe
She had so many sons, she didn’t know what to do.
So she sliced up the shoe, like a loaf, and gave them to her boys
“This is your home, enjoy the trapezoid!”
So, how much living space did each kid have?
Let’s say this is the cross section of the shoe slice:
How to figure out the area?
An area for a trapezoid is the SUM of the two parallel sides, divided by 2, and then MULTIPLIED by the height. The height is a perpendicular line between the two parallel sides.
Here we go:
So let’s give some dimensions to the shoe:
Let’s plug in those values:
Don’t forget this is squared meters since we are considering two dimensions! We multiplied the labels to get meters2
Fun addition: volume!
But the shoe slice also has three dimensions: length!
It is very easy to figure out the volume of a trapezoid! Simply multiply the area by its length!
Since we already know the area, let’s just say that the generous old woman gave each son a slice of shoe that was two meters long. Plug in the numbers again:
*Note that when measuring by volume, we are dealing with three dimensions, so the figure is cubed!
All right, math wizards, here’s a trick question: does a circle have more than 360° in it? The answer: nope. Once you go all the way around a circle, you end up back at the beginning! Pretty easy, right? But that doesn’t mean an angle can’t have more than 360°. Here’s an example:
Say we’re on a Ferris Wheel. If the ride’s any good at all, you just don’t make one trip around and then get off. You keep on going for several rounds, right? For each revolution, we make repeated trips and the degrees we travel doesn’t just start over at 0. We end up with a graph that looks like this:
So what does that have to do with coterminal angles? It’s pretty easy once we think about it. Say we traveled 45° on the initial go around.
Where we started is the Initial side of the angle. Where we stopped is the terminal side of the angle. Neat, huh? If we were going in the opposite direction of the initial side, we would be going in a negative direction. In this case, starting at the initial side and going “backwards” to the terminal side gives us -315°. When we make a complete loop, degrees traveled will be 360° and the negative angle will be 0°. Makes sense, right? But what if we kept going after we completed the loop and landed back at the terminal side? It would look like this:
See how we traveled from in a complete loop from the terminal side? To find out how many degrees we traveled in, simply add 360° to the initial angle! 45°+360°=405° We can say that 45° and 405° are coterminal. But we can also do more! Coterminals can be negative as well. Remember the -315° from going backwards? That angle also shares the same initial and terminal sides. Thus, 45°,-315°,and 405° are all coterminals! *****BASIC RULE: just add or subtract 360° to your initial angle and you can find a coterminal. The possibilities are endless! Take a gander below:
63° and 2223° are coterminals. And we can keep going! How else can we measure coterminals? Well, we don’t have to use degrees. We can use radians. What’s a radian? A radian is the measurement of an angle in a circle where the radius is the same as the angle’s arc. Here’s a radian:
How many radians are in a circle? Well, the formula we use is: there are 2π radians in a complete circle. We divide it like so:
Have a circle is π, and a complete circle would bring us back to 2π. How much is 2π? We know that π=3.14159….and then goes on forever and ever and ever. That makes 2π=6.28318 Since there are 2π in a complete circle, that means there are 6.28318 radians in a circle. That means that 6.28318 radians=360° One rad is 57.296°! Awesomely Rad, I know! The same principle applies to radians as they do degrees. If we need to find a coterminalangle, we can add or subtract 2π! It’s as easy as that. Quick example: we have an angle with radian π/3. What is the coterminal angle?
Just add 2π! π/3+2π→ 2π/6+12π/6→14π/6→7π/3 radians If we want to find more coterminal angles, we can add or subtract 2π! The possibilities are endless! Did you want to convert this measurement to degrees? 7π/3 × (360°)/2π→7/1×120/2→7×60=420° Here are the basic formulas: For coterminals of degrees -> add or subtract 360° For coterminals of radians -> add or subtract 2π Here’s a quick cheat sheet conversion chart from radians to angles:
Congrats! You are now a radical coterminal angle wizard with the help of online learning StudyGate.com platform !
Here’s a secret that math tutors keep to themselves: Sometimes, the fanciest math skills are actually the simplest ones! Like the ultra-mysterious Reference Angle! Sounds pretty impressive, no?
But Reference Angles are actually one of the easiest things to define. Let’s draw a graph and an acute angle (an angle less than 90) and an obtuse angle (an angle that is greater than 90):
The Reference Angles are the measurement of degrees from the shortest distance between the terminal line to the X-axis.
For an acute angle, it is very simple. It is just the degrees.Inn this case, it is 45°.
Fort an obtuse angle, it is a little trickier. The terminal line is actually closest to the x-axis on the OPPOSITE side of the angle. In this case, it would be this:
We know the degree measurement is 70° because the angles that make up a straight line through the y-axis is 180°. We just subtract 110° from 180° to get the REFERENCE ANGLE!
***Please note that even though the Reference Angle is on the negative side of the x-axis, it is always positive.
***Also, unless the terminal side of the angle is on the y-axis to form a right angle, the Reference Angle will always be acute!!!
In other words, Reference Angle ≤ 90°
Here is the largest Reference Angle possible:
It is right angle with a measurement of 90°
***We can use Reference Angles to calculate the functions of angles, like sine, cosine, and tangents. It’s basically a cool shortcut: the sin(70) and sin(110) are the same because they have the same Reference Angle!
Want to see Reference Angles in action??? Check out the link below:
In the whole, wide world of geometry, there are so many shapes and sizes of triangles that you might think they’re all exactly the same, or that they’re all completely different. Not necessarily! Two triangles must share precisely specific traits in order to be deemed similar. If you’re faced with the puzzle of determining if there are two similar triangles, there are three different paths you can take to solve the mystery.
Path 1: Use the Angle-Angle (AA) Method
If you can determine the angle measurement of at least two angles from each triangle, the AA method is your best chance of solving the similarity mystery. If two angles of one triangle are the exact same measurement of the same two angles of the other, the two triangles must be similar. It’s really just that simple!
Path 2: Use the Side-Side-Side (SSS) Method
The SSS method is most helpful when you are given, or can determine, the length of all three sides of both triangles. If all sides on one triangle are specifically proportional to the corresponding sides of the other, then those two triangles must be similar. For example, if one triangle has a hypotenuse of 3 inches, and the second has a hypotenuse of 6 inches, those two sides are proportional because 6 is exactly twice the length of 3. If the other corresponding sides of the triangle are proportional, the triangles are similar and the mystery is solved.
Path 3: Use the Side-Angle-Side (SAS) Method
When you’re faced with two triangles that have at least one given angle measurement, and two given side measurements, the SAS method is your best bet. If the corresponding angles of the triangles have the exact same measurements, and the corresponding sides are proportional to one another, then those two triangles must be similar. Just remember that the measured angle from one triangle must be in exactly the same place as the measured angle from the other, and the same principle applies to the measurements of the two sides as well.
While the industrious student can simply whip out a graph calculator to draw up graphing functions, figuring out functions the old-fashioned way can be pretty keen. With nothing but a pencil and a piece of paper divided into little squares, we can create lines and make ‘em dance to the curve of a slope intercept form.
We all knew math was a cute little number, but here is where we prove it!
Let’s take a simple equation:
y = mx + b
This is a pretty basic format for graphing. We have a Y axis that has a direct relationship with the X axis. The ratio between Y and X is determined by M.
Let’s throw in some numbers and see where they land:
y = 2x + 6
This is as easy as it gets. Let’s whip up a graph:
Let’s start easy, with x = 0.
So we know that when X is at 0, Y is 6. The “2” is the ratio between X and Y. If X was 1, then simply multiply Xx 2 + 6 = 8.
We can create a quick chart between X and Y. Don’t forget that we can go into negative values for X and Y s as well!
Then, plot the points and connect to form a line!
Pretty neat, huh?
****QUICK TIP: go back to the equation
We can break M down to represent the change over Y over the change over X. If it were a fraction, it would look like this: change over Y ÷ change over X
Since 2 is basically 2÷1, we can simply say that for every change in X, we move Y up 2 spaces! Look at the chart and do the math. See how it works?
What is M was a fraction?It still works out:
This means that for every X value we have, Y goes up by . The chart would look like:
Graphs with fraction M look less steep than graphs where M is a whole number.
But where does this get us with graphing functions?
It’s really easy: it’s basically the same thing!
ƒ(x) = x2 + 2
In this case, Y = F(x). Since the equation is squared, there will be some curves as the Y intercept changes from positive to negative. But the procedure is the same.
So let’s throw some X values in and see what we get!
As you can see, we have a curve!
Imagine the possibilities: with numbers, you can draw anything!
When you first see a negative exponent such as 2-2, it can appear confusing. Although you know that 2 squared is 4, what is 2 squared to the -2 power?
The answer is easier than it first appears, and will become simple to understand when you realize that a negative exponent is just a way of telling you to flip the number to become a fraction. Thus, 2-2 is a more streamlined way to tell you the real problem is 1/(22). In other words, the answer to 2-2 is 1/4.
Let’s consider another problem featuring a negative exponent, but with a twist. What would we do if we had 23/2-2?
To make it easy, let’s look at the problem as two different parts: 23 multiplied by 1/2-2. The first part, 2 to the third power, is 2 multiplied by 2 multiplied by 2, or 8. The second part featuring a negative exponent is a bit trickier.
We already know that 2-2 is the equivalent of 1/4, but what happens when a fraction appears as the denominator of another fraction? If you recall from other mathematics assignments, you must reverse the bases and multiply the top number with the bottom number. Therefore, we should reverse the 1/4 to 4/1, or 4. Then, we can multiply it by the top number, which is 1. In other words, 4 multiplied by 1 equals 4.
At this point, we can now put the two parts back together again and solve 23/2-2 as such:
23/2-2 = 8/2-2 = 8 x 4 = 32.
In the world of mathematics, teachers, students, and scientists regularly rely on these types of shortcuts. Negative exponents aren’t meant to complicate matters, but to give you a different way to reorganize your problems. As you become more familiar with solving negative exponent problems, you will find that they are handy and apt in many situations.
As a math student, you probably learned the squares of each number from one to 12 when you were young. For instance, you know that 7 times 7 is 49. Hence, the square root of 49 is obviously 7. But what happens when you are faced with a question like how to find the square root of 12?
You can get to the answer faster than you might have thought by using deductive reasoning and common place formulas.
How to Find Square Root of 12 Step One: Guesstimate Your Answer
Common sense tells you that 12 falls between two other squares, 9 and 16. Nine is the square of 3 and 16 is the square of four. Consequently, you already know that your answer is going to be more than 3 and less than 4.
This knowledge is a practical way to keep your final answer in check. For instance, if you try to find the square root of 12 and end up with a number that begins with 6, you will instantly realize you made an error.
How to Find Square Root of 12 Step Two: Break up the 12
The square root of 12 can be broken into two components because when two square roots are multiplied, the multiplication rules are the same as if they were not square roots. Consequently, the square root of 4 multiplied by the square root of 3 equals the square root of 12. Since you already know that the square root of 4 is 2, you can simplify your problem to 2 multiplied by the square root of 3.
Now, you only have to find the square root of 3 and multiply it by 2, which seems simpler and less wieldy than finding the square root of 12.
How to Find Square Root of 12 Step Three: Use Some Trial and Error
Although there is an algorithmic formula for finding square roots, you can also play around using old-fashioned trial and error. In fact, this is helpful when you are just beginning to encounter square roots because it helps you gain perspective on these fascinating numbers.
For example, you know that the square root of 3 is going to be a number between 1 and 2 multiplied by itself. Therefore, you can start at the midpoint of 1.5. Squared, 1.5 equals 2.25, not quite enough to reach 3. Thus, you know you should go higher. How about 1.7?Squared, 1.7 equals 2.89. That’s much closer to 3.
You can then try to square 1.8, which gives 3.24, more than the prime number 3 you are seeking.
In moments, you have determined that the square root of 3 begins with 1.7. You could add further decimal places if desired, or can stick with 1.7.
How to Find Square Root of 12 Step Four: Finalize Your Answer
For the sake of ease, let’s go with 1.7 as the square root of 3.
Multiply 2 (which was the square root of 4) by 1.7 (our estimated square root of 3) to get 3.4. Then, check the answer by squaring 3.4. You get 11.56, very near to 12. If you want to get nearer, you can quickly add decimal places, changing the estimate to 3.41 (which squares to 11.6), 3.42 (which squares to 11.7), and so on.
How to Find Square Root of 12 Step Five: Getting to the Square Root of 12 or Another Tricky Number
As you can see, it is possible to break down just about any square root of a non-prime through a little math know-how. Test yourself by trying to determine the square root of 20 or 28 for practice. You can always use a square root calculator or online learning siteto check your progress.
Eventually, you can do fast evaluations of all square roots without the need for help, which is a great feeling when you are on your own and facing a test or homework assignment.