Question

@Mertsj
Did I get this problem right?

Use the image below to answer the following question. Find the value of sin x° and cos y°. What relationship do the ratios of sin x° and cos y° share?

Answer 1

In order to find the value of sin x and cos y and its ratio's relationship to each other, we need to first understand what these terms mean. Sin (short for Sine) stands for dividing the opposite side (side opposite of the angle) by the hypotenuse, this is is what it looks like, Opp/Hyp. The Cos (short for Cosine) stands for dividing the adjacent side (side with the angle) by the hypotenuse, this is what it looks like, Adj/Hyp.

As you can see, the hypotenuse is missing, so in order to continue we must find the hypotenuse, and we will do so by using the Pythagorean theorem, this is what it looks like, a2 + b2 = c2. Now I plugged my sides in so it looks likes this:
8(2) + 6(2) = c2

I now multiply and add my sides:
64 + 36 = 100

And finally square 100, which equals 10. I now know that the length of the hypotenuse is 10. Now I can find the Sin and Cos of the triangle.

Sin x: Opp/Hyp
I now go to "x" and find the side that is opposite of it, which is 6, and find the length of the hypotenuse, which is 10. I now set up the fraction of Opp/Hyp, which looks like this: 6/10. Before I finish though, I need to reduce this fraction, and I will do so by 2. This is what it looks like reduced: 3/5. Now we know that the ratio Sin x of this triangle is 3/5.

Cos y: Adj/Hyp
I now go to "y" and find the side which is adjacent of it, which is also 6. I now find the length of the hypotenuse, which is 10. I now set up the fraction of Adj/Hyp, which looks like this 6/10. Before I finish though, I need to reduce this fraction, and I will do so by 2. This is what it looks like reduced: 3/5. Now we know that the ration Cos y of this triangle is 3/5.

As you can see, both ratio's (Sin x and Cos y) are 3/5, which also means that they are equal. This means that the relationship that they share are that they are equal to each other.

Now in order to find the value of Sin x and Cos y we need to apply 3/5 to the inverse of Sin and Cos. This is what it will look like.

Sin x: 3/5 = 0.6 (-1sine) = 36.87

Cos y: 3/5 = 0.6 (-1cos) = 53.13

36.87 + 53.13 = 90

Now we know what the values of Sin x and Cos y are and also know that they make up a right triangle because both angles equaled 90.

Answer 2

@phi
Did I get this right?
Find the value of sin x° and cos y°. What relationship do the ratios of sin x° and cos y° share?

Answer 3

In order to find the value of sin x and cos y and its ratio's relationship to each other, we need to first understand what these terms mean. Sin (short for Sine) stands for dividing the opposite side (side opposite of the angle) by the hypotenuse, this is is what it looks like, Opp/Hyp. The Cos (short for Cosine) stands for dividing the adjacent side (side with the angle) by the hypotenuse, this is what it looks like, Adj/Hyp.

As you can see, the hypotenuse is missing, so in order to continue we must find the hypotenuse, and we will do so by using the Pythagorean theorem, this is what it looks like, a2 + b2 = c2. Now I plugged my sides in so it looks likes this:
8(2) + 6(2) = c2

I now multiply and add my sides:
64 + 36 = 100

And finally square 100, which equals 10. I now know that the length of the hypotenuse is 10. Now I can find the Sin and Cos of the triangle.

Sin x: Opp/Hyp
I now go to "x" and find the side that is opposite of it, which is 6, and find the length of the hypotenuse, which is 10. I now set up the fraction of Opp/Hyp, which looks like this: 6/10. Before I finish though, I need to reduce this fraction, and I will do so by 2. This is what it looks like reduced: 3/5. Now we know that the ratio Sin x of this triangle is 3/5.

Cos y: Adj/Hyp
I now go to "y" and find the side which is adjacent of it, which is also 6. I now find the length of the hypotenuse, which is 10. I now set up the fraction of Adj/Hyp, which looks like this 6/10. Before I finish though, I need to reduce this fraction, and I will do so by 2. This is what it looks like reduced: 3/5. Now we know that the ration Cos y of this triangle is 3/5.

As you can see, both ratio's (Sin x and Cos y) are 3/5, which also means that they are equal. This means that the relationship that they share are that they are equal to each other.

Now in order to find the value of Sin x and Cos y we need to apply 3/5 to the inverse of Sin and Cos. This is what it will look like.

Sin x: 3/5 = 0.6 (-1sine) = 36.87

Cos y: 3/5 = 0.6 (-1cos) = 53.13

36.87 + 53.13 = 90

Now we know what the values of Sin x and Cos y are and also know that they make up a right triangle because both angles equaled 90.

Answer 4

Yes, as I can see it , it is correctly done!

Very good and explanatory, but when you want to write an exponent put a caret (the ^ )
like if you say a^2 it means "a to the second power" which is \(\large\color{black}{ \displaystyle a^2 }\).

Mertsj is offline.

Answer 5

How about this question.
An observer (O) is located 900 feet from a building (B). The observer notices a helicopter (H) flying at a 49° angle of elevation from his line of sight. How high is the helicopter flying over the building? You must show all work and calculations to receive full credit.

Answer 6

In the image above, the height of the helicopter is marked as h, and the distance from the observer to the building is 900 feet. Using the angle of elevation and a trigonometric function, the height of the helicopter can be found.

Looking at the angle of elevation, the known value is the adjacent side (900 feet). The side we are looking for is labeled h (which is the height of the helicopter) which is the opposite side. The trigonometric ratio that relates to the opposite and adjacent sides is the tangent function.

I will now set up the equation to solve this problem:
tan x: Opp/Adj
tan 49: x/900
1.1503684072 = x/900
x = 1035.33

Now we know that the height of the helicopter over the building is 1035.33 feet.

Answer 7

I am re-entreing to get rid of question mark signs.

Answer 8

yes, well done.

Answer 9

you are brilliant! no joke !

Answer 10

How about this one. Had a ton of trouble.

Explain the difference between using the trigonometric ratios (sin, cos, tan) to solve for a missing angle in a right triangle versus using the reciprocal ratios (sec, csc, cot). You must use complete sentences and any evidence needed (such as an example) to prove your point of view.

In order to answer this question, we first need to understand what each of the ratio's are. Here are all of the ratio's.
Sin: Opp/Hyp
Cos: Adj/Hyp
Tan: Adj/Opp

Here are their reciprocals:
Sec: Hyp/Opp
Csc: Hyp/Adj
Cot: Opp/Adj

Now that we understand what the ratio's are, I will do my best to explain the differences between them in using them to solve for a missing angle in a right triangle. In reality, the differences between the two are small but I was able to find (I think) a difference. First, one of the main difference between the normal ratio's and the reciprocals is that in the reciprocal, you divide you answer by 1. For example, when you are trying to find the Sin of 42, we normally just plugged 42 into the calculator and press the Sin button and we get our answer, but for the Csc we not only do this, but we divide our answer by 1. For example, when I put 42 into my calculator and press the Sin button I end up with 0.6691306035, but when using the Csc I would do this and also press the reciprocal button (1/x) and would end up with a new total of 1.4944765498, this is the reciprocal of the result. Also, another main different between the two is that whatever answer the normal function gets, the co-funtion just uses the reciprocal as its answer.

Answer 11

yes, so what happens with the the reciprocals?

Answer 12

when we say that (for example)
\(\large\color{black}{ \displaystyle \sin({\rm C})={\rm Opp}/{\rm Hyp}={\rm c}/{\rm b} }\)
finding the reciprocal (in this Csc(C) ) you are doing sort of the same,
\(\large\color{black}{ \displaystyle \left[~\sin({\rm C})~\right]^{-1}=\left[~{\rm Opp}/{\rm Hyp}~\right]^{-1}=\left[~{\rm c}/{\rm b}~\right]^{-1} }\)
\(\large\color{black}{ \displaystyle \csc({\rm C})={\rm Hyp}/{\rm Opp}={\rm b}/{\rm c} }\)

when finding a missing angle or missing side in a triangle USING "REGULAR" (Sine, Cosine, Tangent) TRIG FUNCTION , YOU ARE FLIPPING EVERYTHING (all fractions) UPSIDE-DOWN.

Answer 13

So really, they only difference between the two is that you are flipping it.

Answer 14

well, as far as equations go yes.
(and of course Sin(C) and Csc(C) will not have the same value , rather) the Sin(C) and Csc(C) will be multiplicative inverses of each other.

Answer 15

and so is for example, Tan(A).


Tan(A) = Opp/Adj = a/c

when you take a reciprocal trig. ratio you get
Cot(A) = Adj/Opp = c/a