Finding the value of theta using the $6$ trigonometric functions on an $xy$ plane.

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Here's a link of what I am trying to learn about.

http://www.webpages.uidaho.edu/learn/math/lessons/lesson03/3_05.htm

Now I have one question. How will I find the value for theta in each function if the hypotenuse has no exact square root and is now a decimal.

Let $(14,5)$ be a point on the terminal side.

$x = 14$

$y = 5$ \begin{align*} r & = \sqrt{14^2+5^2}\\ r & = \sqrt{196+25}\\ r & = \sqrt{221}\\ r & = 14.86 \end{align*}

What do I need to do with the calculator?

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Based on a now deleted comment, it is my understanding that the terminal side of the angle passes through the point $(14, 5)$. You used the Pythagorean Theorem to conclude correctly that $r = \sqrt{221}$.

first-quadrant_angle

Using the trigonometric formulas \begin{align*} \sin\theta & = \frac{y}{r} & \csc\theta & = \frac{r}{y}\\ \cos\theta & = \frac{x}{r} & \sec\theta & = \frac{r}{x}\\ \tan\theta & = \frac{y}{x} & \tan\theta & = \frac{x}{y} \end{align*} with the values $x = 14$, $y = 5$, and $r = \sqrt{221}$ yields the exact values \begin{align*} \sin\theta & = \frac{5}{\sqrt{221}} & \csc\theta & = \frac{\sqrt{221}}{5}\\ \cos\theta & = \frac{14}{\sqrt{221}} & \sec\theta & = \frac{\sqrt{221}}{14}\\ \tan\theta & = \frac{5}{14} & \tan\theta & = \frac{14}{5} \end{align*} Now that we have the exact values, we can plug them into the calculator to obtain the approximations \begin{align*} \sin\theta & \approx 0.34 & \csc\theta & \approx 2.97\\ \cos\theta & \approx 0.94 & \sec\theta & \approx 1.06\\ \tan\theta & \approx 0.36 & \cot\theta & = 2.8 \end{align*} where I have rounded to the nearest hundredth except for $\cot\theta$. The value for $\cot\theta$ is exact.