quick trig question

[yeahbuddy]

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Sorry if i look like a ***** here, but i've always been a bit confused when it comes to trigonometry. what is the best way to solve a problem like this. (topscore cd, QR #2)

if sin(3x-26) = cos(5x-60), find x.

answer is 22


any help is much appreciated!

 

[dane4695]

I wondered about the same question -- when i took the test I just tried #'s and ended up getting it.

 

[japolloniac09]

this is how i'd do it:

we know that sin(3x-26) = cos(5x-60)

we can also say that sin^2(3x-26) + cos^2(5x-60) = 1
since sin2x + cos2x = 1

then substitute for one of them,

cos2(5x-60) + cos2(5x-60) = 1
2cos2(5x-60) = 1
cos2(5x-60) = 1/2

square both sides

cos(5x-60) = 1/sqroot(2) = sqroot(2)/2

and remember that cos45 = sqroot(2)/23

so 5x-60 = 45
x = 22

 

[japolloniac09]

scratch the last 3 in the 3rd to last line

 

[ems5184]

hmm...plugging in numbers i think may be the quickest, i cant think of any easier way at the moment.

but, if you wanted to do it quantitatively, you know that sin(x) = cos(x) when x = 45. so if you set one side of the trig function equal to 45 you get something like: 3x-26 = 45 or 5x-60 = 45. when solving for x you get 23.67 and 21 respectively, and these are both right around 22. im not sure what the answer choices were, but this could work i guess.

im sure there is a way to get the answer directly, but i usually use the guess and check method on these problems....hope this helps yeahbuddy

 

[japolloniac09]

yup that's a lot quicker, as i realized as i was going working through the problem. plugging in numbers would have taken me a longer time than the last two methods, stilll. good call

 

[HBomb]

Plugging #'s may take longer (that's debatable), but here's the beauty of plugging numbers for this question: you don't have to memorize trig identities!!!

 

[ScottW3]

Everything stated so far will work, but there is another very easy way and quicker way (At least as quick as the second method posted).

The is a simple trig property being overlooked. EMS5184 was onto it in his post. Here it is:
When two angles, x and y, are complementary (add up to 90), then:
sin (x) = cos 👍
tan (x) = cot 👍
sec (x) = csc 👍
sin 👍 = cos (x)
tan 👍 = cot (x)
sec 👍 = sec (x)
.
.
.
You get the point. Basically any inverse trig functions are equal when the angle add up to 90.

Knowing this we just solve:
(3x-26) + (5x-60) = 90
x=22
Bam, there you go. Quick, easy, and painless.

I think the trig property I mentioned was in barrons or maybe kaplan. Hope this helps.

Scott

 

[yeahbuddy]

thanks everyone for all the help, much appreciated!