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In trigonometry, the hypotenuse is always the longest side but, the adjacent and opposite sides are not as consistent. When and how do you know which is side is the adjacent vs. opposite in a right triangle? In this diagram, when is seven on the opposite side? When does it become adjacent?

Bonus: To find side AB would I use Tan (19˚): 7/x = 7 tan (19˚)? enter image description here

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    $\begingroup$ This is always with respect to the marked angle given. The way I teach it - start from the vertex where the angle is and "look" out towards a side - that side is the opposite side. $\endgroup$ Nov 4, 2021 at 1:40
  • $\begingroup$ @SeanRoberson Thank you for your explanation. So the 7 is on the opposite side (of the angle), correct? There were times while doing problems, that side was considered adjacent and it threw me off because I did not understand why. $\endgroup$
    – יהודה
    Nov 4, 2021 at 1:45
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    $\begingroup$ In the figure, side $AC$ (of length $7$) is opposite angle $B$ (the $19^\circ$ angle) but adjacent to angle $C$ (the $71^\circ$ angle). Side $AB$ is opposite angle $C$ and is adjacent to angle $B.$ $\endgroup$
    – David K
    Nov 4, 2021 at 3:08
  • $\begingroup$ @DavidK Thank you, Mr. David K. $\endgroup$
    – יהודה
    Nov 4, 2021 at 3:16

1 Answer 1

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You might know that sin(x) , cos(x) and tan(x) are what's known as trigonometric functions. FUNCTIONS , and as what's common with any function f(x), they have an argument , which is the value you replace x with, and thus get an output, with respect to the provided input.

Similarly, these trigonometric functions are blind to the triangle you are applying them to, they are simply providing an output with respect to the argument you provide it. You might have learnt the "soh cah toa" as a trick to remember what ratios they represent, so applying this to your triangle, if I feed the argument B to tan(x) , it will spit out a certain value, which will be equal to the ratio represented tan(19°); that will be 'opposite to B' divided by 'adjacent to B'. (Opposite side to an angle is the side not connected to it, and since hypotenuse remains fixed, adjacent side is the last of the three)

What you have done seems to be correct, except that x=7cot(19°).

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  • $\begingroup$ Thank you. When exactly do you use the opposite (Cosecant, Secant, and Cotangent)? How do you know when to use them? I am more familiar with the inverse function than I am with the opposites. $\endgroup$
    – יהודה
    Nov 4, 2021 at 2:05
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    $\begingroup$ There is virtually no purpose these serve other than to make writing the statements a bit cleaner. For example, 1/(sin(x).cos(x)) could be written as sec(x).cosec(x). I hope you can see that there's no statement in cosec, sec and cot that couldn't be expressed in terms of sin, cos and tan. Knowing when to use them would completely depend on whether the question demands it to be represented in a certain way. $\endgroup$ Nov 4, 2021 at 2:20
  • $\begingroup$ Understood, thank you $\endgroup$
    – יהודה
    Nov 4, 2021 at 2:21

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