Finding trigonometric ratios of complementary angles (worked examples)

which of these options is equivalent to cosecant eighty degrees okay so I have some options here and one of them is equal to because it says choose one answer one of them is equal to this and the rest are not I have to find which one that is now I have a clue here evaluating trigonometric ratios of complementary angles some kind of looking at some complementary relationship so okay how do I write here okay so I need to know what cosecant theta is equal to I know that cosecant of any angle cosecant of any angle theta will be equal to secant of 90 minus theta right this is just a property that I'm aware of because I know what these ratios really mean and if ever you are a little bit doubtful you can just draw the triangle and verify that your that what you've written down here is correct because you can look at it okay there is let's say there is theta here then cosecant is basically hypotenuse by opposite and for the angle 90 minus theta that's same ratio co-ceo hypotenuse my opposite would have become for this angle or hypotenuse by adjacent which is what we call secant theta so for 90 minus theta are the same ratio will become secant 90 minus theta and that will be equal to cosecant of this angle cosecant of theta can always verify this and now that you know this after this it simply becomes plugging the value of theta and finding what it should be so secant of 90 minus 80 in this case and you get secant 10 degrees secant of 90 second of 10 degrees and there is an option that says secant 10 degrees so let's select that and let's check that's right keep up the good work now let's let's go to the next question cuz he can 75 degrees equals secant of X which x value will make the equation true okay it's in effect just asking me secant of what is equal to cosecant of 75 once again I know my relationship between cosecant and secant so cosecant of 75 degrees equals that's right we just did it so you know that it's secant of 90 minus 75 degrees so 90 minus 75 equals secant of 15 right so 15 plus 75 will be 90 so that's correct so secant of 15 degrees let's verify yeh I never get tired of this string sound it it always brings brings joy next question so here we say which of the options is equivalent to cos of 1 degrees now once again I know that just like cosecant and secant share a relationship sine M cos 2 so sine of theta will be equal to cos of 90 minus theta and once again you can draw the diagram to verify this is true so cos of 1 will be equal to sine of 89 which is 90 minus 1 let me just write that cos of 90 minus 1 which is cos 89 shouldn't skip steps you know and Orsi keep going or see how we answer the question ok I'll see how you answer the question so we can use this identity to find the value of x which is cos theta equals 90 minus theta sine of 90 minus theta which is what we did so we follow the same method that khanacademy solution has as well so finally we have one question involving cot and tan so cot 23 equals tan of would once again cot of 23 will be equal to tan of 90 minus 23 now this is because you can draw the triangle again you can verify this all of these are complementary relationships so that's going to be equal to tan of 90 minus 20 is 70 minus 3 67 degrees and with that I would have hopefully got all of the questions correct [Music]