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I have a right angled triangle question and have through my working figure out that the angles are $90^\circ$, $67^\circ$, and $23^\circ$. I have one side opposite $23^\circ$ which is $13$cm. I need to find the side length opposite the right angle.

I've tried sine rule and I get a negative number.

$$\sin \frac{23}{13} = \sin \frac{90}{CB}$$

Through rearranging (possibly incorrect?)

$$CB = \frac{\sin 90}{ \frac{\sin 23 }{13}} ≈ -19$$ when typed in my calculator.

What am I doing wrong?

Any help is much appreciated; sorry if the question is very basic.

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    $\begingroup$ but you have given all angles? $\endgroup$ – Dr. Sonnhard Graubner Jun 5 '17 at 12:18
  • $\begingroup$ It might be usefull if you added a picture of your triangle with these values marked on it, so it is easier for us to follow what you are trying to do. After all "a picture speaks a thousand words" $\endgroup$ – lioness99a Jun 5 '17 at 12:18
  • $\begingroup$ There is no angle defined as opposite of another angle in a triangle. $\endgroup$ – Zubzub Jun 5 '17 at 12:19
  • $\begingroup$ i think you mean the side length opposite the right angle? $\endgroup$ – Dr. Sonnhard Graubner Jun 5 '17 at 12:20
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    $\begingroup$ I think your calculator is using rad $\endgroup$ – stity Jun 5 '17 at 12:20
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This is a more complete solution to the ones mentioned above:

I think that a more straightforward way would be to use the sine here. As @zubzub mentioned, the side opposite a right angle is a hypotenuse (by definition). If we call the hypotenuse $h$, since sine = opposite/hypotenuse, $\sin 23º = 13/h$, so by rearrangement $(\sin 23º) (h) = 13$, or $h = 13/ \sin 23º$.

In future problems like this, keep 3 things in mind:

1) Make sure you know which units you are using. Degrees often have whole numbers and are much larger (from 0 to 360) than radians, which are often in multiples of $\pi$ (such as $5/3 \pi$).

2) Use the appropriate rule: the sine rule and cosine rule apply for non-right angled triangles, while for right angled triangles the basic trigonometric functions (sine, cosine, tangent and their inverses) are more practical, since they can be computed by definition.

3) Make sure to check your order of operations: $\sin(23/13)$ is a very different thing to $\sin(23)/13$! Check your calculator to see whether it follows the order of operations or not (most scientific calculators do), and add brackets in the right places when you are not sure whether to place them.

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  • $\begingroup$ The sine rule is used when the triangle doesn't have a right angle, so the basic trigonometric functions such as sine and cosine are of more practical use in this question. $\endgroup$ – Toby Mak Jun 5 '17 at 12:32
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Since $\sin(90^\circ)=1$, all you need to compute is $\frac{13}{\sin(23^\circ)}\simeq33.27$.

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  • $\begingroup$ Could you also tell me what step of my working was wrong? $\endgroup$ – Charlie Jun 5 '17 at 12:34
  • $\begingroup$ As others said, you probably computed the sin of $13\operatorname{rad}$, instead of the sine of $13^\circ$. $\endgroup$ – José Carlos Santos Jun 5 '17 at 12:39
  • $\begingroup$ Do you mean 23°? $\endgroup$ – Charlie Jun 5 '17 at 12:40
  • $\begingroup$ Yes, that is what I meant. Sorry about that. $\endgroup$ – José Carlos Santos Jun 5 '17 at 12:40
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if so then you have $$\sin(23^{\circ})=\frac{13cm}{c}$$ therefore $$c=\frac{13cm}{\sin(23^{\circ})}$$

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