What is this angle in a right triangle with sides of length 5, 12, and 13?

Question

How do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13.

Thanks

Answer 1

Use the fact that your triangle is a right triangle, denote your angle $\theta$ (the angle you need, assuming it's the angle that leg b forms with the hypotenuse), and use the definition of $\sin \theta$ or $\cos \theta$ or $\tan \theta$ as each is defined in terms of the lengths of the sides you are given.

Any one of these trig functions of $\theta$ can be used, as you are given the lengths of the side opposite the angle (a), adjacent to the angle (b), and the hypotenuse (h).

Example: $$\sin \theta = \frac{\text{opposite side length }}{\text{hypotenuse side length}} = \frac{a}{h} = \frac{5}{13}$$

$$\small \left(\cos\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13},\;\;\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}\right)$$

Then solve for $\theta$, knowing $$\theta = \sin^{-1}\left(\dfrac {5}{13}\right) = \arcsin\left(\dfrac{5}{13}\right)$$

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The above relationships between the lengths of the sides of a right triangle and $\cos\theta, \sin\theta, \tan\theta$ should become "second nature" to you.

There are mnemonics that can help you memorize/remember this. For example, the mnemonic I learned was:

$$\text{"The Old Arab Sat On His Camel And Hiccuped"}\iff $$ $$ \text{ Tan = Opposite/Adjacent, Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse}.$$

There are other such mnemonics that exist, as well, and perhaps others can post them as comments below!

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Resources

See Right-Angled Trigonometry, a pdf document you can open and/or download that covers the concepts need to solve problems like the one you posted.

See this link to better understand how to solve these kinds of questions. E.g., To answer this question, it suffices to know the lengths of two sides of the right triangle.

Answer 2

So you're referring to the angle opposite leg a, which we'll call angle A. We have

$$\sin A=\frac ac=\frac5{13}$$

To find the measurement of the angle, we need to use the inverse sine function, yielding

$$A=\sin^{-1}\frac5{13}$$

I don't think there's any way to simplify that, but if you want an approximate value, you can probably compute it with a scientific calculator.

Answer 3

Hint: $\sin(A)=\frac{5}{13}$, $\cos(A)=\frac{12}{13}$, and $\tan(A)=\frac{5}{12}$. $A$ can be found using the inverse of $\sin$, $\cos$, or $\tan$.