Theta is the symbol used for the unknown angle in a triangle. When working with right angle triangles – triangles with one 90 degree angle – you can use Soh Cah Toa to help you find the unknown angle.

Before learning how to find the missing angle right away, there’s a few steps to understand.

First, label your triangle. This will let you know which Trigonometry function to use when calculating Θ.

There’s the hypotenuse – the longest length of the right angle triangle, the opposite – the angle opposite to the Θ or the angle you are looking for, and the last length left is called the adjacent.

In order to find Θ, you need to know at least two lengths.  This can help you apply Soh Cah Toa.

Soh Cah Toa is an acronym to help you find with trig function you should use in order to find Θ. The first letter of each three is the trig function; S = sine, C = Cosine, T = tangent. The next two letters are the side lengths and their order; o = opposite, a = adjacent, h = hypotenuse. So; Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

When you know two lengths and you know their side names, you can find the function you need to solve for Θ. For example, if you know the length of the opposite and hypotenuse, you can use sine to find the angle.

When solving for an angle, you use the inverse of the function.

sin(Θ)= 20/29 

Θ = sin^–1(20/29)

Θ = 44 degrees. (rounded to the nearest ten)

When finding a length when you know your angle, you can use one of the functions. For example if we know the opposite and the angle and are trying to find the hypotenuse, we can use sine again. x = the unknown length (hypotenuse)

sin(44 degrees) = 20/x

20 ÷ sin(44 degrees) = x

29 (rounded to the nearest ten) = x

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I found trig to be a very easy subject of math. As long as you were able to identify the lengths of the triangle, there was an acronym to follow after and formulas that made finding the number simple.

A right pyramid is a pyramid with apex(the vertex at the tip of the pyramid) and with triangular sides.

To find the volume, you need to know what the dimensions of the base and the height.

Example

To find the surface area of a right pyramid, you need to know the slant height and the base length.

Example

If you do not know either of these dimensions, you can use The Pythagorean Theorem to help you find it.

Example

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A right cone is a 3D shape with a circular base and vertex directly above the base.

To find the volume of cone, you need to know the dimensions of the base and the height.

Example

To find the surface area of a right cone, you need to know the radius and the slant height.

Example

If you don’t know either of these dimensions, you can use The Pythagorean Theorem to help you find it.

Example

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With using The Pythagorean Theorem, it became a lot easier to find the volumes and surface areas. As long as I knew all the dimensions and inputted it correctly into my calculator, I got the right answer.

We had to find an equation for the surface area to a cylinder that equals between 520 -540 cm squared. 
Then we had to create the cylinder. 

The advantage of converting within the Metric System is that all the units are related to each other by a factor of ten.  You can use, to convert between units, a metric unit number line. Each movement on this number line presents a power of ten. 

When converting between different units, we need to find how many positions on the line we move. The middle represents the main unit; Meters, Grams, Liters. If the unit chart is true; 1 meter to a centimeter is two steps to the right. Meaning 1 meter is equal to 10 to the power of 2 centimeters, or 100 centimeters. The power is the amount of steps moved; if moving towards to right, the power will be positive; if moving towards the left, the power will be negative.

Example of converting in the SI system

If we want to convert 15 700 cL to ML, we need to find where cL and ML are on the number line.

Since it is 8 steps to the left, this means the difference is 10 to the power of -8; 1 cL = 0.000000008ML

Now that we found the conversion, we can apply it to the number we are converting. You want to multiply the number that you want to convert (15 700 cL) with the difference in units (0.00000001 over 1cL).

Note: you want the cL in the denominator so the units can cancel out.

When cancelled out, you can multiply the numbers to find the answer.

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I found that this lesson can either be very complicated or very simple. I learned that the key is that you have to write every step down so you don’t confuse yourself and also to remember what unit you are converting to what, so
you don’t mix them around and start from the wrong unit. It is a challenging because of the amount of steps you must remember and follow.

 

A base that is raised to a negative exponent has an integral exponent. 

The general rule for when a base is raised to a negative exponent This is because exponents have a pattern of being divisible by the base, so if continued past exponent 0, you begin to get fractions

You flip it from the numerator to the denominator or the denominator to the numerator. This flip makes the exponent go from a negative exponent to a positive exponent. 

This law remains the same for evaluating different expression

When there are numbers with positive exponents and negative exponents, only the bases with the negative exponents need to move.

The general rule of fraction base to a negative exponent is

This is because 

Evaluating an expression with a fraction for a base and a negative exponent 

You flip the fraction to make the outer negative exponent, a positive exponent. Then proceed to continue simplifying

This unit was very challenging, it called for many different exponent laws for different situations. Once it is explained, though, and I figured out when to use which exponent law, I got the hang of it. When many different of these exponent laws are seen in the same expression, it may seem over whelming to simplify or evaluate, but if you follow through with each law, it become quite easy.

Radical 

A radical is an expression in the form ofAn example of a radical is It includes a index, radicand, and radical symbol, making all together a radical. In this case the index is 5 and the radicand is 125. 

Note:

If the index is not written, as in square root, it is assumed to be 2.

The index is the number of times the radical must be multiplied by itself to equal the radicand.

An entire radical is when the number is entirely under the root symbol or the radical sign. Whereas a mixed radical isn’t. A mixed radical is a form of simplifying an entire radical.

 

 

 

Converting an entire radical to a mixed radical

To convert or simplify an entire radical, you need to find two factors of the number, one being a perfect root. For example,

4 being the perfect root in this case. Then each of those factors become a radical, leaving you with radical that isn’t a whole number (because it is not perfect, and becomes a decimal if rooted). You are then left with a mixed radical. 

If you wish to simplify it futher, you need to repeat the process by finding another perfect root that is a factor of the radicand. 

Converting a mixed radical back to an entire radical

You need to put the number on the outside of the radical sing back under the sign. To do this, you use the index as an exponent for that number. Than multiply it by the radicand. You end up with an entire radical.

 

This week we learned about Radicals, mixed radicals to entire radicals, and entire radicals to mixed, no matter what the index is. One challenge with this lesson is remembering all the steps. In order to make sure that your end result is correct, you need to make sure you don’t forget any step and write everything down so you do not get confused. If you forget to write out each factor as a radical, you can forget to find the roots, therefore leaving you confused. Once I got the hang of it, it became a very easy lesson/subject.

GCF

GCF or Greatest Common Factor is the highest number that can divide exactly into a set of numbers. (For example: 7 is the GCF of 14 and 49, 7 divide into 14 twice and divides into 49 seven times.)

Some numbers are easy enough to do the math mentally (For example: 3 and 9, you can quickly see that the highest factor that goes into both those numbers is 3), but when there are higher numbers or more than a set of two numbers, you can use prime factorization in order to help you find the GCF.

The First Step

Find the prime factorization of the given numbers. (Prime Factorization – Prime factors of a number, the sum is equal to the number)

The Second Step

Find the prime factors in common, also choose the smallest exponent between the common prime factors. 

The Last Step

Multiply all the common factors and the smallest exponents in order to find the GCF. 

(The same steps no matter how high the numbers or how many numbers you are working with.)

LCM

LCM or Lowest Common Multiple is the lowest number that is a multiple of a set of numbers. (For Example: 12 is the lowest common multiple of 2, 3, and 4.)

Again, for finding LCM, you could probably do the math mentally when the numbers are small like 2, 3, and 4. But for higher numbers, instead of listing out all the multiples of each number and finding which is the lowest multiple in common, we can use prime factorization.

The First Step

Find the prime factors of each number.

The Last Step

Take all the prime factors and their highest exponent. Multiply to find the LCM.

This week, we learned about prime factors, prime factorization, and how to apply prime factorization when trying to find the GCF and LCM. One challenge with this lesson is that there is a lot of mental math and math without a calculator. You need to figure out a lot when it comes to factors especially, like doing the math on whether a number divides exactly into another number when trying to find the prime factors. It is a very good thing to learn because it trains our minds and helps us learn how to do math without a calculator.

 

 

When you know the two legs of a right angled triangle, but don’t know the hypotenuse, you can always use the Pythagorean Theorem. Square each leg, and add them together. You will find the answer is the hypotenuse squared. To find the final answer find the square root.  If you know the hypotenuse and one of the legs, but not the other one, you can use the Pythagorean Theorem backward. Square the numbers you know. Subtract one of the legs from the hypotenuse, you will end up with the answer for one of the legs squared. You need to find the square root and you end up with the final answers.