Week 15 – Precalc 11 – Caleb’s Blog

Merry Christmas and Happy Holidays!

It’s the last week of school before the Winter break to celebrate Christmas and New Year’s! Week 15 was a mixed bag. We were trying to remember last year’s introduction to trigonometry as well as learning new concepts of trigonometry. The trigonometric ratios were taught to us as an acronym “SOH CAH TOA”.

sin θ = $\frac{Opposite}{Hypotenuse}$

cos θ = $\frac{Adjacent}{Hypotenuse}$

tan θ = $\frac{Opposite}{Adjacent}$

Now, these same primary trigonometric ratios are defined differently on an x- and y-coordinate grid graph.

sin θ = $\frac{y}{r}$

cos θ = $\frac{x}{r}$

tan θ = $\frac{y}{x}$

where $r$ = $\sqrt{x^2+y^2}$

A good example of using these ratios is question 7 on page 412.

7a) A support cable is anchored 15m from the base of a pole and is attached to the pole 10m above the ground. Determine the length of the cable.

Using the Pythagorean Theorem, we can calculate the length of the cable (or hypotenuse).

$x^2 + y^2 = r^2$

$(15)^2 + (10)^2 = r^2$

(225) + (100) = $r^2$

(325) = $r^2$

$\sqrt{(325)}$ = $\sqrt{r^2}$

$5\sqrt{13}$ = $r$

18.027756 = $r$

The length of the cable is approximately 18m.

7b) What angle does the cable make with the ground?
With the information given, we can use either 2 trigonometric ratios to solve this angle. The first ratio is the COSINE.

cos θ = $\frac{Adjacent}{Hypotenuse}$

cos θ = $\frac{15}{18}$

θ = $cos^{-1}$ $(\frac{15}{18})$

θ = 33.557°

The other ratio is TANGENT.

tan θ = $\frac{Opposite}{Adjacent}$

tan θ = $\frac{10}{15}$

θ = $tan^{-1}$ $(\frac{10}{15})$

θ = 33.690°

The cable makes a 34° angle with the ground (according to both answers).

Lastly, we learned to use “Special Triangles” to easily identify the basic Sine, Cosine, and Tangent ratios.

The above diagrams show the trigonometric ratio for certain angles. That list is organized in the table below. Because these ratios are used frequently in calculus, we were told to memorize these! Oh boy!!

Angles	sin ⁡θ	cos⁡ θ	tan θ
0°	$\frac{0}{1}$	$\frac{1}{1}$	$\frac{0}{1}$
30°	$\frac{1}{2}$	$\frac{\sqrt {3}}{2}$	$\frac{1}{\sqrt {3}}$
45°	$\frac{1}{\sqrt {2}}$	$\frac{1}{\sqrt {2}}$	$\frac{1}{1}$
60°	$\frac{\sqrt {3}}{2}$	$\frac{1}{1}$	$\frac{\sqrt {3}}{1}$
90°	$\frac{1}{1}$	$\frac{0}{1}$	$\frac{1}{0}$ (undefined)
180°	$\frac{0}{1}$	$\frac{-1}{1}$	$\frac{0}{-1}$
270°	$\frac{-1}{1}$	$\frac{0}{1}$	$\frac{-1}{0}$ (undefined)
360°	$\frac{0}{1}$	$\frac{1}{1}$	$\frac{0}{1}$

Week 15 – Precalc 11

Leave a Reply Cancel reply