“Is it possible to construct a triangle with the side lengths of 7, 8, and 15?”

In the usual sense of things (Euclidean Geometry), no.

You could have a ‘degenerate triangle‘, but that’s not exactly a triangle in the way we usually think about a triangle.

8 + 7 = 15.

So basically you want the sum of the two smaller sides to be larger (even slightly larger) than the largest side to get a triangle that isn’t degenerate, a triangle in the way we usually think about triangles.

triangle_8_7_15

Fractional Exponents, Logarithms, Quartic and Cubic Equations

We started by looking at some current homework which involved factoring. For most of the problems they started with a quartic or cubic equation and then factored into a quadratic and one or two other terms.

Cubic equations can have three solutions, quartic equations can have four.

Talked a little about one application of imaginary numbers since we saw complex numbers in some solutions. And went into more detail about how to do calculations with them.

Talked about fractional exponents for a bit and how to work with them. Also about logarithms. The base being 10 if no other base is shown.

Fractional exponents are almost always easier to work with than ‘cube roots’, ‘fourth roots’ etc. Square roots aren’t too bad but still can be harder to manipulate in some cases.

Last problem we worked on was a three dimensional volume of a cone with the top cut off. Looked at the application of similar triangles and the proportionality of the sides.

“How many radians are in 180 degrees?”

180 degrees is the angle of half a circle.

360 degrees is the angle of a full circle.

In radians, a full circle is 2π.

So half of that is a half circle and half of 2π is π.

If 180 degrees is the angle for a half circle and π is the angle for half a circle then both are measures of the same angle.

180 degrees in radians

How to find the length of a square given the perimeter?

If you have the perimeter of a square, the outside is made of four equal sides.

square diagonal 1
After that I would draw in the diagonal and see that it forms a right triangle, you can also notice that it forms a 45 45 90 right triangle. From there, you can either use the Pythagorean theorem, take cos(45 degrees), or remember the relationship between the sides for a 45 45 90 triangle.
square diagonal 2

Three options.

Geometry Problem with Regular Hexagon

Since the side lengths of the hexagon are all equal, it is a ‘regular’ hexagon. Therefore all the interior angles of the hexagon are equal.

To find the sum of the interior angles of the regular hexagon, you can use the formula

(n-2)180 degrees

Where n is the number of sides, six in this case.

The angle you’re trying to find is labeled in red, circled by black.

You could find it by subtracting the right angle from the interior angle.

hexagon angles

Or you could look at the angle on the left, which is an interior angle and recognize that it is the largest angle in an isosceles triangle.

From there you could subtract the interior angle from the total of 180 degrees for a triangle and then divide by two to find one of the two congruent angles.

“How can I find the height of a cone by only knowing the radius and length of the side?”

The cross section of a cone is a right triangle, if you know that you can use the Pythagorean theorem.

volume of a right circular cone radius

“How do I find the slant height of a cone with only the radius and curved surface area given?”

The formula for the lateral surface area of a cone is the radius multiplied by the slant height multiplied by pi.

LSA = rlπ

You can move parts of that equation around depending on the information you have.

slant height lateral surface area

 

“What is the exact value of sin−1(sin(4π7)) ?”

The expected answer would be 4π/7, but that is not the answer since 4π/7 is outside the acceptable range of [−π/2,π/2] for the arcsin function. If you took the arcsin of the sin of an angle within the range of [−π/2,π/2] then it would be the angle that you took the sin of.

However, the angle in this problem is outside that range.

Visually

The first thing I did was place the angle given in radians on a graph.

arcsin and sin of angle in radians

Here’s how I thought about it,

3.5π/7 = π/2 (same position as 90 degrees)

4π/7 is slightly larger than that. It’s the angle shown with the blue line on my diagram. Probably slightly larger than it should be, but qualitatively accurate for this problem. Notice that the side opposite the angle (in purple) is positive.

4π/7 is 0.5π/7 to the left of π/2. 3π/7 is 0.5π/7 to the right of π/2.

There are two possible angles that will give us the same result if we plug them into the sin function.

The angle with the orange line is 3π/7.

If you take the sin of 4π/7 or 3π/7, you get the same result (with the same sign) since the opposite side of the triangle is positive in either case. The hypotenuse is always positive for this situation.

If you took the sin of -4π/7, you would get a negative number. That’s shown off to the right side on the bottom in light blue.

Functions, domain/range

The arcsin function expects a value between -1 and 1. sin has a domain of [−π/2,π/2] and range is [−1,1]. Arcsin has the domain and range switched. If it’s outside those bounds, the function and its opposite are not one to one.

Equivalent values

sin(3π/7) = sin(4π/7)

And 3π/7 is within the proper range of [−π/2,π/2] for the arcsin function.

Therefore if you take the arcsin of sin(3π/7) or sin(4π/7), it will give you the one value that is within the range of [−π/2,π/2].

And that is 3π/7

“What is an isosceles triangle?”

An isosceles triangle is a triangle with two sides that are equal (congruent).

An equilateral (three equal sides) triangle is also isosceles, but an isosceles triangle is not necessarily equilateral.

Here are a few examples of isosceles triangles.

iscosceles triangles

Volume of a Cone Formula

volume of a right circular cone

Assuming it is a right circular cone, here’s the equation for the volume and two ways to think about it.

For the cone, you have the height, radius, and slant height.
pi r^2 is the area of a circle, so knowing that makes this calculation simpler.

 

You can multiply the area of the base (the circle) by the height.