if cos 0= 0.5 find csc (0- pi/2)
cosθ = 1/2
make a sketch of the corresponding triangle, x=1, r=2
x^2 + y^2 = r^2
y^2 = 3
y = ±√3
so sinθ = √3/2 in quadrant I and sinθ = -θ 3/2 in quad IV
sin(θ -π/2) = -sin(π/2 - θ) = -cosθ , because of the complementary angle property
thus csc(θ - π/2) = - secθ = -2
or, even simpler
since cosθ = .5 , θ = 60°
csc(60-90)
= csc(-30°)
= -csc(30°) = -1/(1/2) = -2
To find csc(x) where cos(0) = 0.5, we first need to find sin(x).
Given that cos(x) = 0.5, we know that cos(x) = adjacent/hypotenuse. We can create a right triangle with the adjacent side as 0.5 and the hypotenuse as 1 (since cosine is the ratio of the adjacent side to the hypotenuse).
Using the Pythagorean theorem, we can find the opposite side:
(a^2) + (b^2) = (c^2)
0.5^2 + b^2 = 1^2
0.25 + b^2 = 1
b^2 = 0.75
b = sqrt(0.75)
b = 0.866
So, sin(x) = opposite/hypotenuse = 0.866/1 = 0.866.
Now, to find csc(x) = 1/sin(x), we have:
csc(x) = 1/0.866
csc(x) ≈ 1.155.
Therefore, csc(0 - π/2) is approximately equal to 1.155.
To find the value of csc(0 - π/2), where cos(0) = 0.5, we need to use trigonometric identities and the reciprocal relationship between sine and cosecant.
Cosecant (csc) is the reciprocal of sine (sin), which means csc(x) = 1 / sin(x). Therefore, we need to find the sine of (0 - π/2) to calculate csc(0 - π/2).
To find the value of sin(0 - π/2), we can use the identity sin(-x) = -sin(x). In this case, x = π/2, so sin(0 - π/2) = -sin(π/2).
We know that sin (π/2) = 1, so -sin(π/2) = -1.
Now that we have found the value of -sin(π/2), we can calculate csc(0 - π/2) using the reciprocal relationship:
csc(0 - π/2) = 1 / (-1) = -1
Therefore, csc(0 - π/2) = -1.