Math Snap

STEP 1

What is this asking?
We need to find the sine, cosine, and tangent of angles $\theta$ and $\phi$ in a right triangle, knowing two of its sides.
Watch out!
Don't mix up which sides are opposite and adjacent to which angle!
Also, we'll need to find the hypotenuse first.

STEP 2

1. Find the hypotenuse
2. Calculate sin θ, cos θ, and tan θ
3. Calculate sin φ, cos φ, and tan φ

STEP 3

Alright, let's do this!
We've got a right triangle, so we can use the Pythagorean theorem: $a^2 + b^2 = c^2$, where a and b are the legs and c is the hypotenuse.
In our triangle, $a = 4$ and $b = 16$.

STEP 4

Let's plug in the values: $4^2 + 16^2 = c^2$.
This simplifies to $16 + 256 = c^2$, so $c^2 = 272$.

STEP 5

Taking the square root of both sides gives us $c = \sqrt{272}$.
We can simplify this!
Since $272 = 16 \cdot 17$, we have $c = \sqrt{16 \cdot 17} = \sqrt{16} \cdot \sqrt{17} = 4\sqrt{17}$.
So, our hypotenuse is $4\sqrt{17}$!

STEP 6

Remember SOH CAH TOA?
For angle $\theta$, the opposite side is 16, the adjacent side is 4, and the hypotenuse is $4\sqrt{17}$.

STEP 7

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{4\sqrt{17}} = \frac{4}{\sqrt{17}}$.
We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{17}$ to get $\sin \theta = \frac{4\sqrt{17}}{17}$.

STEP 8

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{4\sqrt{17}} = \frac{1}{\sqrt{17}}$.
Rationalizing gives us $\cos \theta = \frac{\sqrt{17}}{17}$.

STEP 9

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{4} = 4$.

STEP 10

Now for angle $\phi$!
The opposite side is 4, the adjacent side is 16, and the hypotenuse is still $4\sqrt{17}$.

STEP 11

$\sin \phi = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{4\sqrt{17}} = \frac{1}{\sqrt{17}}$.
Rationalizing gives us $\sin \phi = \frac{\sqrt{17}}{17}$.

STEP 12

$\cos \phi = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{16}{4\sqrt{17}} = \frac{4}{\sqrt{17}}$.
Rationalizing gives us $\cos \phi = \frac{4\sqrt{17}}{17}$.

STEP 13

$\tan \phi = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{16} = \frac{1}{4}$.

(a) $\sin \theta = \frac{4\sqrt{17}}{17}$
(b) $\sin \phi = \frac{\sqrt{17}}{17}$
(c) $\cos \theta = \frac{\sqrt{17}}{17}$
(d) $\cos \phi = \frac{4\sqrt{17}}{17}$
(e) $\tan \theta = 4$
(f) $\tan \phi = \frac{1}{4}$