Sine this, cosine that....???

[Nismoboy]

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I have a question which I couldn't figure out even after looking at the explanation. Here it is:

If sin (theta)= 1/2 cos(theta), and 0 is less than or equal to (theta) which is less than or equal to Pi/2, the value of 1/2 sin (theta) is:

A) 0.22
B) 0.25
C) 0.45
D) 0.5
E) 0.75

Just in case you're wondering or couldn't understand the problem the way I wrote it, it's on page 108 of the Kaplan DAT Review Notes (equivalent of the Kaplan "Blue Book"). Thanks.

 

[coodoo]

i dont have the book, so dont get mad if this is wrong😀

sin(theta) = 1/2cos(theta) or 2sin(theta) = cos(theta)

now square both sides:

4sin^2(theta) = cos^2(theta)

we know that cos^2(theta) is equal to (1-sin^2(theta)) **TRIG ULTRA RULE sin^2(theta)+cos^2(theta) = 1**

so we have: 4sin^2(theta) = 1-sin^2(theta) or

5sin^2(theta) = 1 and now

sin^2(theta) = 1/5 : now take the square root of both sides and get

sin (theta) = 1/sqrt of 5....now multiply the denominator and numberator by sqrt of 5 and you end up with **sqrt of 5/5**...now take 1/2 of this since you want 1/2 sin(theta) and you end up with sqrt 5/10 = .22. I'm pretty sure they would tell you what the sqrt of 5 is on the real dat

 

[coodoo]

if they dont just remember that the sqrt of 5 is less than 2.5, which leave A as the only logical answer b/c 2.5/10 =.25 and there is only one answer that is smaller than this

 

[ScottW3]

That is a pretty good method, but a slightly easier way (in my opinion) at arriving to the answer is:

sin (theta) = (1/2) * cos (theta) so,

sin (theta)---1
---------- = ---- and,
cos (theta)---2

tan (theta) = (1/2)


because tan (theta) = (opposite / adjacent) we know these two sides (1 and 2 respectively)

We can find the 3rd side by 2^2 + 1^2 = hypotenuse^2
-------------------------------sqrt(5) = hypotenuse

So then sin (theta) = opposite / hypotenuse, sin (theta) = 1/ sqrt(5)

Which brings you to the final answer as stated above,
(1/2)(sqrt(5)/5) = sqrt(5)/10

I just find this to be a little quicker, and any time I can save on QR is good. 🙂

Hope it helps...

 

[coodoo]

That is a pretty good method, but a slightly easier way (in my opinion) at arriving to the answer is:

sin (theta) = (1/2) * cos (theta) so,

sin (theta)---1
---------- = ---- and,
cos (theta)---2

tan (theta) = (1/2)


because tan (theta) = (opposite / adjacent) we know these two sides (1 and 2 respectively)

We can find the 3rd side by 2^2 + 1^2 = hypotenuse^2
-------------------------------sqrt(5) = hypotenuse

So then sin (theta) = opposite / hypotenuse, sin (theta) = 1/ sqrt(5)

Which brings you to the final answer as stated above,
(1/2)(sqrt(5)/5) = sqrt(5)/10

I just find this to be a little quicker, and any time I can save on QR is good. 🙂

Hope it helps...

great logic...indeed a much faster way 👍

 

[jjdaddy]

*shrug* You don't have to know square root of 5 in order to solve this one.

Remember that sine(theta) and cos(theta) where 0 <= theta <= pi/2
are 0 to 1.

If 1/2 sin (theta) is greater than .25, then sin(theta) is greater than .5.
Since cos(theta) = 2 *sin(theta), cos(theta) is greater than 1. From this
reasoning, you can eliminate choices C) D) E)

If 1/2 sin (theta) = .25 then sin(theta) = .5 and cos(theta) which is twice the sin(theta) is 1. But, when cos(theta) is 1, theta is 0.
And sin(0) = 0. sin(theta) can't be both 0 and .5!!! From these two opposing statement, you can eliminate choice B).

Thus, only possible answer remaining is A)

I have a question which I couldn't figure out even after looking at the explanation. Here it is:

If sin (theta)= 1/2 cos(theta), and 0 is less than or equal to (theta) which is less than or equal to Pi/2, the value of 1/2 sin (theta) is:

A) 0.22
B) 0.25
C) 0.45
D) 0.5
E) 0.75

Just in case you're wondering or couldn't understand the problem the way I wrote it, it's on page 108 of the Kaplan DAT Review Notes (equivalent of the Kaplan "Blue Book"). Thanks.