4. Find the exact value for sin(x+y) if sinx=-4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant.
5. Find the exact value for cos 165degrees using the half-angle identity.
1. Solve: 2 cos^2x - 3 cosx + 1 = 0 for 0 less than or equal to x <2pi.
2. Solve: 2 sinx - 1 = 0 for 0degrees less than or equal to x <360degrees.
3. Solve: sin^2x = cos^2x for 0degrees is less than or equal to x < 360degrees.
4. Solve sinx - 2sinx cosx = 0 for 0 is less than or equal to x < 2pi.
#4
sin(x+y) = sinx cosy + cosx siny
so we need the cosx and the siny
from sinx = -4/5 we know we are dealing with the 3,4,5 right-angled triangle, so in the fourth quadrant cosx = 3/5
from cosy = 15/17 we know we are dealing with the 8,15,17 right-angled triangle
so in the fourth quadrant, siny = -8/17
then
sin(x+y) = sinx cosy + cosx siny
= (-4/5)(15/17) + (3/5)(-8/17)
= -84/85
cos 2A = 2cos^2 A - 1
let A = 165, then 2A = 330
so let's find cos 330
cos(330)
= cos(360-30)
= cos360 cos30 + sin360 sin30
= (1)(√3/2) + (0)(1/2) = √3/2
then √3/2 = 2cos^2 165 - 1
(√3 + 2)/4 = cos^2 165
cos 165 = -√(√3 + 2))/2
(algebraically, our answer would have been ± , but I picked the negative answer since 165 is in the second quadrant, and the cosine is negative in the second quadrant)
I will give you hints for the rest,
you do them, and let me know what you get.
#1 factor it as
(2cosx - 1)(cosx -1) = 0
so cosx - 1/2 or cosx = 1
take it from there, you should get 3 answers.
#2, the easiest one
take the 1 to the other side, then divide by 2,
#3, take √ of both sides to get
sin 2x = ± cos 2x
sin2x/cos2x = ± 1
tan 2x = ± 1 etc
#4 factor out a sinx etc
4. To find the exact value for sin(x+y) when sinx = -4/5 and cosy = 15/17, we can use the sum formula for sine, which states that sin(x + y) = sin x * cos y + cos x * sin y.
Since x and y are both in the fourth quadrant, sin x and sin y will both be negative. Therefore, sin x = -4/5 and sin y = -√(1 - cos^2 y) = -√(1 - (15/17)^2).
We can substitute these values into the formula to get:
sin(x + y) = (-4/5)(15/17) + (-√(1 - (15/17)^2))(15/17)
Simplifying this expression gives us the exact value for sin(x + y).
5. To find the exact value for cos 165 degrees using the half-angle identity, we need to use the identity cos^2 (θ/2) = (1 + cos θ)/2.
Let's start by converting 165 degrees to radians. 165 degrees = (165 * π) / 180 = 11π/12.
Now we can use the half-angle identity: cos^2 (11π/24) = (1 + cos (11π/12))/2.
To find the exact value, we need to determine cos (11π/12). Since this angle is not one of the commonly known values on the unit circle, we'll use the reference angle (π/12) to find its value.
Since π/12 corresponds to a 30-degree angle, we know that cos (30 degrees) = (1/2). Therefore, cos (π/12) = (1/2).
Substituting this into the half-angle identity, we get: cos^2 (11π/24) = (1 + (1/2))/2.
Simplify this expression to find the exact value for cos (11π/24).
1. To solve the equation 2cos^2x - 3cosx + 1 = 0 for 0 ≤ x < 2π, we can use factoring or the quadratic formula.
One way to solve this equation is by factoring:
2cos^2x - 3cosx + 1 = 0
(2cosx - 1)(cosx - 1) = 0
We set each factor equal to zero:
2cosx - 1 = 0
cosx = 1/2
cosx - 1 = 0
cosx = 1
Solving for x gives us two possible solutions:
cosx = 1/2 --> x = π/3 or x = (5π)/3
cosx = 1 --> x = 0
Therefore, the solutions to the equation are x = 0, x = π/3, and x = (5π)/3.
2. To solve the equation 2sinx - 1 = 0 for 0 ≤ x < 360 degrees, we can isolate the sine term and then solve for x.
First, add 1 to both sides of the equation:
2sinx = 1
Next, divide both sides by 2:
sinx = 1/2
To find the angle whose sine is 1/2, we can use the inverse sine function or reference angles. Since 1/2 corresponds to a 30-degree angle, we know that sin(30 degrees) = 1/2.
Therefore, the solutions to the equation are x = 30 degrees and x = 150 degrees.
3. To solve the equation sin^2x = cos^2x for 0 degrees ≤ x < 360 degrees, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Given the equation sin^2(x) = cos^2(x), we can substitute it into the Pythagorean identity:
(sin^2x) + (sin^2x) = 1
2sin^2x = 1
sin^2x = 1/2
Taking the square root of both sides gives us two possible solutions:
sinx = ±√(1/2)
Using the reference angle, we find that sin(45 degrees) = √(1/2).
Therefore, the solutions to the equation are x = 45 degrees and x = 135 degrees.
4. To solve the equation sinx - 2sinx cosx = 0 for 0 ≤ x < 2π, we can factor out sinx from the equation:
sinx(1 - 2cosx) = 0
Now we have two possibilities:
1. sinx = 0, which means x = 0 or x = π.
2. 1 - 2cosx = 0, which gives us 2cosx = 1, cosx = 1/2, and x = π/3 or x = (5π)/3.
Therefore, the solutions to the equation are x = 0, x = π, x = π/3, and x = (5π)/3.