Question

Please help?

Answer 1

1. `theta = 22.6^@`
2. `theta = 17.1^@`
3. `theta = 48.2^@`
4. `theta = 50.0^@`

Explanation:

It's time to use our new friend, SOHCAHTOA!

As a refresher, this stands for:

• `"sine" = "opposite" / "hypotenuse" " "(S=O / H)`
• `"cosine" = "adjacent" / "hypotenuse"" "(C = A / H)`
• `"tangent" = "opposite" / "adjacent"" "(T = O / A)`

What does it mean? Well, in any right triangle, there are 3 sides.
Relative to one of the angles (other than the `90^@` one), these are called

• the opposite side (the one opposite our angle),
• the hypotenuse (the one opposite the right angle), and
• the adjacent side (the one next to our angle, but not the hypotenuse).

For example, in question (1), the angle is `theta` (theta), with opposite side `bar(BC),` hypotenuse `bar(AB),` and adjacent side `bar(AC).` (As shorthand, each side's name is usually the lowercase letter of its opposite angle, e.g side `bar(AB)` would be called side `c`.)

What we notice is that, if we scale a right triangle up or down, the angles all remain the same, but so does the ratio of any two sides. For example, if a right triangle with sides 3, 4, and 5 doubles in size, its sides become 6, 8, and 10, but the ratio `3/5` matches the ratio `6/10.`

So for any right triangle, scaling the whole triangle doesn't change either the angles or the ratio of any two sides. It's like they're linked; an angle determines a ratio, and vice versa. How cool is that?! It's so cool, we write a relation between the angle `theta` and each of the 3 side pairs.

The pair "opposite to hypotenuse" gets the name sine, and we define it as `sin theta = "opposite"/"hypotenuse".` Similarly, we define the cosine of an angle as `cos theta = "adjacent"/"hypotenuse",` and tangent is `tan theta="opposite"/"adjacent".`

Finally, the answer!

In question 1, we're given an angle `theta` and its adjacent side (12) and the hypotenuse (13). The trigonometric ratio that uses these sides is cosine, so to solve for `theta`, we write:

`cos theta = "adj"/"hyp"=12/13`

So we know the cosine of `theta` is `12/13`. To find `theta` itself, we need to "undo" this cosine action. There should be a button on your calculator that looks like `cos`, with a `cos^"-1"` above it. If we have an angle and want the `"adj"/"hyp"` ratio, we use `cos.` To go the other way, we use `cos^"-1".`

Using this "inverse cosine" function, we get

`costheta = 12/13`
`=>theta = cos^"-1"(12/13)~~22.6^@`

I'll leave the calculation of the other `theta`'s as an exercise. As a hint, question 2 will use tangent (and thus the `tan^"-1"` button on your calculator.)