## Angles of Elevation & Shadows

If you have the angle of elevation from the sun (or another light source) and you have the height of a person, you can calculate the length of their shadow.

There are two similar triangles, the smaller triangle with the height of the person and the larger triangle with the height of the sun. They share an angle theta.
Sometimes setting up the problem and getting the concept is the hardest part.

## Why the Tangent Function Can be Greater Than 1, But the Sine and Cosine Functions Cannot

Regardless of the angle for which you evaluate tangent, cosine, or sine, you can always think of it as the ratio of two sides of a right triangle. See the orange angle (quadrant 1), green angle (quadrant 2), purple angle (quadrant 3), and red angle (quadrant 4) in the diagram.

For the tangent function, you can divide the opposite side by the adjacent side.

For the sine function, you can divide the opposite side by the hypotenuse.

For the cosine function, you can divide the adjacent side by the hypotenuse.

Three sides for Right Triangle

H: hypotenuse, O: opposite, A: Adjacent

For the three sides of a triangle, the two sides that are not the hypotenuse (opposite & adjacent) will never be larger than the hypotenuse. They can be equal to the hypotenuse, so you can get results of the sine and cosine functions being equal to one.

However, the opposite side of a triangle can definitely be larger than the adjacent side of a triangle.

In the triangle with green letters O, H, A the opposite and adjacent sides are similar with the opposite side being a bit larger.

In the triangle with blue letters O, H, A, the opposite side is significantly larger than the adjacent side.

In the third orange triangle, the opposite side is smaller than the adjacent side.

## How Does the Equation for Arc Length Work?

I would suggest thinking about how it works visually/graphically. It should make sense that way.

The formula stands for the arc length being equal to the radius multiplied by the angle (in radians).

The equation can be written as s = r θ

Whole Circle & Half a Circle

If you found the circumference of a circle, it would be 2 π r.

If you found half the circumference of a circle, it would be π r.

The angle included in a whole circle is 2 π, the angle included in half a circle is π.

You multiply the angle (in radians) by the radius to get the arc length for both a circle and half a circle.

And it turns out you multiply the angle (in radians) by the radius to get the arc length for any arc length with an angle.

If the angle is 45 degrees, that’s π/4 radians. So you multiply the angle by the radius to get the arc length. It’s a quarter of a circle.

Regardless of the angle, to get an arc length, you multiply the angle (in radians) by the radius.

# What are the measures of the four angles of the parallelogram?

Here’s a diagram with the information given, plus the necessary statement that the sum of the angles of a four sided figure is 360 degrees. For a triangle it’s 180.
(n-2)180 for the sum of the interior angles where n is the number of sides of the figure.

## The longest side if a triangle is nine meters longer than the shortest side…….

The longest side if a triangle is nine meters longer than the shortest side.
The other side is twice the length of the shortest side.
The perimeter of the triangle is 25 m.
Solve for all three sides

Probably good to start by sketching a diagram.

Then set up an algebraic expression for each statement in the problem.

Translate the words into separate equations (using all of the information given)

1. The longest side if a triangle is six units more than the shortest side

2. The other is twice the length of the shortest side

3. P=25

The part that I see come up more than once in the description is the shortest side.

The length of the longest side is described in terms of the shortest side and the length of the second longest side is also described in terms of the shortest side.

You could potentially use more variables, but it’s better to use the least number of variables possible to reduce the work necessary.

Therefore I’ll describe the sides of the triangle (from shortest to longest) as

1. x
2. 2x
3. x + 9

Then we can combine all the equations to solve because we know the perimeter.

x + 2x + x + 9 = 25

The next step is to combine like terms

4x + 9 = 25

After that we will subtract nine from both sides of the equation to isolate the variable.

4x = 16

Then divide each side by four to get x.

x = 16/4 = 4

Once we have x, we can substitute into the three equations to get all three side lengths

1. x
2. 2x
3. x + 9

1. 4
2. 8
3. 13