Trigonometry Questions
1. Solve the equation sin2x+1=-2sinx for 0≤x≤2π
2. Solve the equation 7sin2x-4sin2x/cosx=-1 where 0≤x≤2π
3. Determine the exact value of cos2(theta) when tan(theta)=3/4 and π<theta<3π/2
4. The average number of customers, c, at a 24 hour sandwich shop per hour is modeled roughly by the equation c(h) = -5cos( π/12h) + 12, with h=0 representing midnight. What time of the day is peak business?
will you take a look at the fun that even Wolfram had in solving the equation the way you typed it ???
http://www.wolframalpha.com/input/?i=solve+sin2x%2B1%3D-2sinx
I have a feeling that you meant:
sin^2 x + 1 = -2sinx
then
sin^2 x + 2sinx + 1 = 0
(sinx + 1)^2 = 0
sinx +1 = 0
sinx = -1
and x = 270° or 3π/2
confirm the typing of the others before I attempt them
thanks! i forgot to include the exponent. I re-typed them here:
2. Solve the equation 7sin^2x-4sin2x/cosx=-1 where 0≤x≤2π
3. Determine the exact value of cos2(theta) when tan(theta)=3/4 and π<theta<3π/2
4. The average number of customers, c, at a 24 hour sandwich shop per hour is modeled roughly by the equation c(h) = -5cos( π/12h) + 12, with h=0 representing midnight. What time of the day is peak business?
2. 7sin^2x - 4sin2x/cosx = -1
7sin^2 x - 4(2sinxcosx)/cosx = -1
7sin^2x - 8sinx + 1 = 0
(7sinx - 1)(sinx - 1) = 0
sinx = 1/7 or sinx = 1
if sinx = 1
x = 90° or π/2
if sinx = 1/7, then
x = aprr 8.2° or 171.8°
3. cos 2Ø
= cos^2 Ø - sin^2 Ø
given: tan Ø = 3/4 (I recognize the 3-4-5 right-angled triangle
so sinØ = 3/5 and cosØ = 4/5
then cos 2Ø
= cos^2 Ø - sin^ Ø
= 16/25 - 9/25 = 7/25
4. c(h) = -5cos(π/12 h) + 12
c ' (h) = 5(π/12)sin(π/12 h)
= 0 for a max of c(h)
sin(π/12 h) = 0
π/12 h = 0 or π
1/12 h = 1
h = 12
12 hours from midnight would bring you to 12:00 noon
so the peak business is at noon
1. To solve the equation sin^2(x) + 1 = -2sin(x) for 0≤x≤2π, we can follow these steps:
Step 1: Let's rearrange the equation to have all terms on one side:
sin^2(x) + 2sin(x) + 1 = 0
Step 2: Notice that this equation can be factored as a quadratic equation:
(sin(x) + 1)^2 = 0
Step 3: Taking the square root of both sides, we get:
sin(x) + 1 = 0
Step 4: Now, solve for sin(x):
sin(x) = -1
Step 5: Looking at the unit circle, we can see that sin(x) = -1 for x = 3π/2.
Therefore, the solution to the equation sin^2(x) + 1 = -2sin(x) for 0≤x≤2π is x = 3π/2.
2. To solve the equation (7sin^2(x) - 4sin^2(x))/cos(x) = -1 for 0≤x≤2π, we can follow these steps:
Step 1: Let's simplify the left-hand side of the equation:
(7sin^2(x) - 4sin^2(x))/cos(x) = 3sin^2(x)/cos(x) = 3tan^2(x)
Step 2: Set this expression equal to -1:
3tan^2(x) = -1
Step 3: Divide both sides by 3:
tan^2(x) = -1/3
Step 4: Take the square root of both sides, while considering the range of x:
tan(x) = ±√(-1/3)
Step 5: Since the tangent function is negative in the third and fourth quadrants, we consider only the negative square root:
tan(x) = -√(1/3)
Step 6: To find the value of x in the desired range, we can use the inverse tangent function:
x = arctan(-√(1/3))
Therefore, the solution to the equation (7sin^2(x) - 4sin^2(x))/cos(x) = -1 for 0≤x≤2π is x = arctan(-√(1/3)).
3. To determine the exact value of cos^2(θ) when tan(θ) = 3/4 and π < θ < 3π/2, we can use the Pythagorean identity and the given information:
Step 1: Recall the Pythagorean identity: tan^2(θ) + 1 = sec^2(θ)
Step 2: Substitute the given value of tan(θ):
(3/4)^2 + 1 = sec^2(θ)
Step 3: Simplify:
9/16 + 1 = sec^2(θ)
25/16 = sec^2(θ)
Step 4: Take the square root of both sides, considering the range of θ:
sec(θ) = ±√(25/16)
Step 5: Since sec(θ) is positive in the second and fourth quadrants, we consider only the positive square root:
sec(θ) = 5/4
Step 6: Recall that: cos(θ) = 1/sec(θ)
Step 7: Substitute the value of sec(θ):
cos(θ) = 1/(5/4)
cos(θ) = 4/5
Step 8: Square the value of cos(θ) to find cos^2(θ):
cos^2(θ) = (4/5)^2
cos^2(θ) = 16/25
Therefore, the exact value of cos^2(θ) when tan(θ) = 3/4 and π < θ < 3π/2 is 16/25.
4. To determine the time of day at which the sandwich shop has peak business, we need to find the value of h that maximizes the function c(h) = -5cos(π/12h) + 12. Here's how we can do that:
Step 1: Recall that the cosine function oscillates between -1 and 1.
Step 2: We want to find the maximum value of c(h), which is when the cosine term is -1.
Step 3: Set cos(π/12h) = -1 and solve for h:
-1 = -5cos(π/12h)
cos(π/12h) = 1/5
Step 4: Use the inverse cosine function to find the value of π/12h:
π/12h = arccos(1/5)
Step 5: Solve for h by multiplying both sides of the equation by 12/π:
h = 12/π * arccos(1/5)
Step 6: Calculate the value of h using the arccos function and the given formula.
Therefore, the time of day at which the sandwich shop has peak business is h = 12/π * arccos(1/5).