1. Use half-angle identity to find the exact value of cos165.
MY ANSWER: (-1/2)sqrt(2+sqrt(3))
2. Solve 2 sin x + sqrt(3) < 0 for 0<= x<2pi.
MY ANSWER: (4pi/3)< x < (5pi/3)
3.Write the equation 2x+ 3y-5=0 in normal form?
(-2sqrt(13)/13)x - (3sqrt(13)/13)y + 5sqrt(13)/13) = 0
your first one cannot be right since your result < - 1,
and the cosine and sine of any angle cannot be greater than 1 or less than -1.
there are several ways to split up 165
1. 165 = 180 - 15
2. 165 = 90 + 75, but 75 = 45 + 30
unless I misses an obvious combination, it looks like you have do this in 2 steps
third question is correct, if you follow the formula
(don't know why anybody would want to write an equation which looks very neat and clean in such a complex looking form)
To find the exact value of cos165, we can use the half-angle identity for cosine. The formula states:
cos(2θ) = 2cos²(θ) - 1
To apply this, let's consider half of angle 165, which is 82.5 degrees or π/6 radians.
So, cos(165) = cos(2 * 82.5) = 2cos²(82.5) - 1
Now, we need to find the value of cos(82.5) using trigonometric identities.
We'll use the double-angle identity for cosine:
cos(2θ) = cos²(θ) - sin²(θ)
In this case, θ = 41.25 degrees or π/12 radians.
So, cos(82.5) = cos(2 * 41.25) = cos²(41.25) - sin²(41.25)
Using the half-angle identity for sine, we have:
sin(θ/2) = √[(1 - cosθ) / 2]
sin(41.25) = √[(1 - cos82.5) / 2]
Solving for cos82.5:
1 - cos²(82.5) = 2sin²(41.25)
cos²(82.5) = 1 − 2sin²(41.25)
cos(82.5) = −√[1 − 2sin²(41.25)]
Now, we can substitute this value back into the initial equation to find cos(165):
cos(165) = 2cos²(82.5) - 1
= 2[−√(1 − 2sin²(41.25))]² - 1
= (-1/2)√(2 + √3)
Therefore, the exact value of cos165 is (-1/2)√(2 + √3).
-----------------------------------------------------------------------
To solve the inequality 2sin(x) + √3 < 0 for 0 ≤ x < 2π, we need to isolate sin(x).
First, we subtract √3 from both sides of the inequality:
2sin(x) < -√3
Next, we divide by 2 to isolate sin(x):
sin(x) < -√3/2
Now, we need to find the values of x in the range 0 ≤ x < 2π where sin(x) is less than -√3/2.
To do this, we look at the unit circle and identify the quadrants where sin(x) is negative and greater than -√3/2. These are the quadrants where x has a range on the interval [2π/3, 4π/3].
So, the solution to the inequality is 2π/3 < x < 4π/3, where 0 ≤ x < 2π.
Thus, (4π/3) < x < (5π/3) is the solution to 2sin(x) + √3 < 0 for 0 ≤ x < 2π.
-----------------------------------------------------------------------
To write the equation 2x + 3y - 5 = 0 in normal form, we need to ensure that the coefficients of x, y, and the constant term are normalized (i.e., they have equal denominators).
First, let's multiply the equation by the least common multiple of the denominators (in this case, 13). Doing so, we get:
13(2x) + 13(3y) - 13(5) = 0
Simplifying:
26x + 39y - 65 = 0
Now, we can rearrange the equation to have the coefficient of the x-term as the first term, the coefficient of the y-term as the second term, and the constant term on the right side. Doing so, we have:
26x + 39y = 65
To normalize the equation, we can divide all terms by the greatest common divisor of the coefficients (in this case, 13). Dividing by 13, we get:
(26/13)x + (39/13)y = 65/13
Simplifying further:
2x/13 + 3y/13 = 5
To make the coefficients integers, we can multiply through by 13, which gives us:
2x + 3y = 5
In normal form, the equation 2x + 3y - 5 = 0 becomes 2x + 3y = 5.