Can someone please check my answers and help me with the last question!
1. Solve sin2xcos2x = 4sin2x on the interval [0, 2pi]
0, pi, 2pi?
2. Exact value of sin(pi/12) - sin(5pi/12)
root3/4?
3. Using factorial notation, 0! = 1
False?
4. Find the area of the following triangle. a=4.8inches, b=6.3inches, c=7.5inches. Round answer to nearest tenth
area is 1/2bh...so can I make any side the base/height?
#1
sin2xcos2x = 4sin2x
sin2x(cos2x-4) = 0
sin2x = 0 or cos2x = 4
so, 2x = 0,π,2π,3π,4π
x = 0,π/2,π,3π/2,2π
naturally, cos2x=4 has no solutions.
Again, you found solutions for 2x in [0,2π], but you want all values for x in [0,2π]
#2 Since sin u − sin v = 2 sin(½(u−v)) cos(½(u+v)),
sin(pi/12) - sin(5pi/12)
Using the half-angle formula,
= √((1 - cos pi/6)/2)-√((1 + cos pi/6)/2)
= 1/√2 (√(1-√3/2)-√(1+√3/2))
= 1/√2 (√((2-√3)/2) - √((2+√3)/2))
= 1/√2 (1/2) (√(4-2√3)-√(4+2√3))
= 1/2√2 ((√3-1)-(√3+1))
= 1/2√2 (-2)
= -1/√2
Or, using the sum/difference formulas,
sin u − sin v = 2 sin(½(u−v)) cos(½(u+v))
sin(pi/12) - sin(5pi/12)
2sin(-pi/6)cos(pi/4)
= 2(-1/2)(1/√2)
= -1/√2
#3 True. It is defined that way.
4. yes, any side can be the base.
so how do you get the height?
You might consider Heron's formula.
I asked my teacher for answer choices for #2 and she said -root2/2, root2/2, or -root3/4...
you said -1root2
as you surely know by now
1/√2 = 1/√2 * √2/√2 = √2/2
c'mon, man.
Gotcha!
I get 15.03 for the area...is that correct?
Looks good to me.
Did you crank that out manually? I wen to
wolframalpha.com and typed in
√(s(s-a)(s-b)(s-c)) where a=4.8 and b=6.3 and c=7.5 and s=(a+b+c)/2
Yes I got the formula and plugged the values in :)
Sure! Let's go through each question and explain how to find the answers.
1. Solve sin2x * cos2x = 4sin2x on the interval [0, 2pi]
To solve this equation, let's start by factoring out sin2x:
sin2x * (cos2x - 4) = 0
Now we set each factor equal to zero:
sin2x = 0 -> x = 0, pi, 2pi
cos2x - 4 = 0 -> cos2x = 4 (which is not possible since cosine ranges from -1 to 1)
Therefore, the only solutions are x = 0, pi, and 2pi. So your answer of 0, pi, and 2pi is correct.
2. Exact value of sin(pi/12) - sin(5pi/12)
To find the exact value, we can use the half-angle identity for sine:
sin(x/2) = sqrt((1 - cos(x)) / 2)
sin(pi/12) = sin((pi/6)/2) = sqrt((1 - cos(pi/6)) / 2)
= sqrt((1 - sqrt(3)/2) / 2)
= sqrt((2 - sqrt(3))/4)
sin(5pi/12) = sin((5pi/6)/2) = sqrt((1 - cos(5pi/6)) / 2)
= sqrt((1 + sqrt(3)/2) / 2)
= sqrt((2 + sqrt(3))/4)
Now we can subtract these values:
sin(pi/12) - sin(5pi/12) = sqrt((2 - sqrt(3))/4) - sqrt((2 + sqrt(3))/4)
The exact value cannot be simplified further, so your answer of sqrt(3)/4 is incorrect.
3. Using factorial notation, 0! = 1
This statement is actually true. By definition, 0! (read as "zero factorial") is equal to 1. It is a convention agreed upon by mathematicians.
4. Find the area of the following triangle. a = 4.8 inches, b = 6.3 inches, c = 7.5 inches. Round answer to the nearest tenth.
To find the area of a triangle, we can use Heron's formula:
Area = sqrt(s(s-a)(s-b)(s-c))
where s is the semiperimeter of the triangle, given by:
s = (a + b + c) / 2
In this case:
s = (4.8 + 6.3 + 7.5) / 2
= 9.3
Now we can substitute this value in the area formula:
Area = sqrt(9.3(9.3-4.8)(9.3-6.3)(9.3-7.5))
= sqrt(9.3 * 4.5 * 3 * 1.8)
= sqrt(241.425)
Rounding to the nearest tenth, we get:
Area ≈ 15.5 square inches
So the area of the triangle is approximately 15.5 square inches.