Trigonometric Function?

Question

In a play area a children's slide is constructed as shown here : http://i16.tinypic.com/4mtuhkk.jpg
(angle at end of slide/ground is 30°, angle at end of stairs/ground is 45°, and the length of the support beam is 2m)

a)What is the EXACT length of the stairs?
b)What is the EXACT length of the slide?

Please show you trig work/steps. thanks =)

Answer 1

Important facts to know about "special triangles":
1) In a 45-45-90 triangle, the two sides are equal (since its an isoceles triangle). This tells you that if the height of the slide is 2 meters, then the distance from the bottom of the stairs to the support is also 2 meters. Using the Pythagorean theorem, base^2 + height^2 = stairs^2
Thus 2^2 + 2^2 = stairs^2 so
8 = stairs^2 so stairs = 2*sqrt(2) = approx 2.828
2) In a 30-60-90 triangle, he shortest side is one half the longest side (the hypotenuse (in this case the slide itself). The other side (by the Pythagoren Theorem) is short side * sqrt(3).
Thus, the shortest side is the support, which equals 2. The hypotenuse must then be twice this, or 4.

Answer 2

Because the ladder side of the slide makes a 45 degree angle with the ground, and the beam makes a 90 degree angle, that means the other angle must be equal to 45 degrees (to make 180).

On the other side, the beam also makes a 90 degree angle with the ground, and since we know the slide makes a 30 degree angle with the ground, the top of the slide and the beam must form a 60 degree angle.

Now that we know all that, we need to find a way to relate the stuff we know and the stuff we don't know.

Using Sine(X) = Opposite / Hyponteuse, we can say that on the left side, the Sin(30) = 2 / h. If you solve for h, you get 2/sin(30) = h = 4.. The length of the slide is exactly 4 meters.

You can do the same thing on the other side. The Sin(45) = 2 / h. If you solve for h again, you will find that the length of the latter approximately 2.8284 meters, but the exact answer would to leave it as 2/sin(45).

Answer 3

Notice that you have two right triangles here. The support beam forms a right angle with the ground; and the slide is the hypotenuse of one of the triangles; while the stairs form the hypotenuse of the other triangle.

From the definition of sine, we know:

(2m)/(length of slide) = sin(30) [for one triangle]

(2m)/(length of stairs) = sin(45) [for the other triangle]

Now you can use algebra to solve for (length of slide) and (length of stairs).

Now, note that the question asks for an EXACT answer; this means they don't want you to just punch in "sin(30)" and "sin(45)" into your calculator; that will only give you an approximate answer accurate to 8 or 10 digits. Instead, you need to find an exact expression for sin(30) and sin(45). (Hint: each involves a square root.) I'm not going to give away the answer; that's something you can look up in your book.

Answer 4

the three elementary trig purposes are sin, cos, and tan. think of a few staggering triangle, with angles A, B = ninety tiers, and C. opposite those angles are the aspects a, b, and c. Then sin(A) = a / b And cos(A) = c / b And tan(A) = a / c The "area" of a function is the numbers which you will positioned into the function. For sin, cos, and tan, you could put in any selection (any perspective). The area is -infinity to + infinity. The "selection" is the numbers which you get out of the function. sin(...) is often interior the selection -a million to +a million. cos(...) is often interior the selection -a million to +a million. tan(...) is often interior the selection -infinity to +infinity.