Find sin x/2, cos x/2, and tan x/2
from the given information.
cos x=− 4/5, 180° < x < 270°
sin(x/2=
cos(x/2)=
tan(x/2)=
To find the values of sin(x/2), cos(x/2), and tan(x/2), we can use the half-angle formulas.
Given that cos(x) = -4/5, we can determine sin(x) using the Pythagorean Identity:
sin(x) = √(1 - cos^2(x))
= √(1 - (-4/5)^2)
= √(1 - 16/25)
= √(25/25 - 16/25)
= √(9/25)
= 3/5
Since x is in the third quadrant (180° < x < 270°), sin(x) is negative. Thus, sin(x) = -3/5.
Now, we can find sin(x/2) using the half-angle formula:
sin(x/2) = ±√((1 - cos(x))/2)
= ±√((1 - (-4/5))/2)
= ±√((1 + 4/5)/2)
= ±√((9/5)/2)
= ±√(9/10)
= ±3/√10
= ±(3√10)/10
Since sin(x/2) will have the same sign as sin(x), we have sin(x/2) = -(3√10)/10
Similarly, we can find cos(x/2) and tan(x/2) using the half-angle formulas:
cos(x/2) = ±√((1 + cos(x))/2)
= ±√((1 + (-4/5))/2)
= ±√((1 - 4/5)/2)
= ±√((1/5)/2)
= ±√(1/10)
= ±1/√10
= ±√10/10
tan(x/2) = sin(x/2)/cos(x/2)
= (-(3√10)/10) / (±√10/10)
= -(3√10)/10 * 10/√10
= -3√10
Therefore, sin(x/2) = -(3√10)/10, cos(x/2) = √10/10, and tan(x/2) = -3√10.
To find sin(x/2), cos(x/2), and tan(x/2) from the given information, first, we need to use the half-angle identity formulas for sine, cosine, and tangent.
The half-angle identities are:
sin(x/2) = ± √((1 - cos(x)) / 2)
cos(x/2) = ± √((1 + cos(x)) / 2)
tan(x/2) = sin(x/2) / cos(x/2)
Given that cos(x) = -4/5, we can substitute this value into the half-angle identity formulas.
sin(x/2) = ± √((1 - cos(x)) / 2)
sin(x/2) = ± √((1 - (-4/5)) / 2)
sin(x/2) = ± √((1 + 4/5) / 2)
sin(x/2) = ± √(9/10)
Since the angle x lies in the third quadrant (180° < x < 270°), the sine value will be negative.
Therefore, sin(x/2) = -√(9/10) = -3√(10)/10
cos(x/2) = ± √((1 + cos(x)) / 2)
cos(x/2) = ± √((1 + (-4/5)) / 2)
cos(x/2) = ± √((1 - 4/5) / 2)
cos(x/2) = ± √(1/10)
Since the angle x lies in the third quadrant (180° < x < 270°), the cosine value will be negative.
Therefore, cos(x/2) = -√(1/10) = -√10/10 = -√10/10
tan(x/2) = sin(x/2) / cos(x/2)
tan(x/2) = (-3√(10)/10) / (-√10/10)
tan(x/2) = -3√(10)/√10
tan(x/2) = -3
Therefore, sin(x/2) = -3√(10)/10, cos(x/2) = -√10/10, and tan(x/2) = -3.