cosx= -8/17, 180<x<270
find;
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 need to use the half-angle formulas for sine, cosine, and tangent.
The half-angle formula for sine is:
sin(x/2) = ±√[(1 - cos(x)) / 2]
The half-angle formula for cosine is:
cos(x/2) = ±√[(1 + cos(x)) / 2]
The half-angle formula for tangent is:
tan(x/2) = sin(x) / (1 + cos(x))
Let's substitute the given value of cos(x) into these formulas.
Given: cos(x) = -8/17
Using the half-angle formulas, we can find:
sin(x/2) = ±√[(1 - cos(x)) / 2]
= ±√[(1 - (-8/17)) / 2]
= ±√[(17 + 8) / (2*17)]
= ±√[25/34]
= ±5/√34
(Note: We take the negative square root since x is in the third quadrant, where sine is negative.)
cos(x/2) = ±√[(1 + cos(x)) / 2]
= ±√[(1 + (-8/17)) / 2]
= ±√[(17 - 8) / (2*17)]
= ±√[9/34]
= ±3/√34
(Note: We take the positive square root since cos(x/2) is always positive in the given interval.)
tan(x/2) = sin(x) / (1 + cos(x))
= (-√34/5) / (1 - 8/17)
= (-√34/5) / (17 - 8)/17
= (-√34/5) * (17/9)
= (-17√34 / 45)
So, the values are:
sin(x/2) = -5/√34
cos(x/2) = 3/√34
tan(x/2) = -17√34/45
To find the values of sin(x/2), cos(x/2), and tan(x/2), we can use the half-angle identities. The half-angle identities are given by:
sin(x/2) = ±√((1 - cos(x))/2)
cos(x/2) = ±√((1 + cos(x))/2)
tan(x/2) = sin(x/2) / cos(x/2)
Let's calculate these values step-by-step.
Given information: cos(x) = -8/17, 180° < x < 270°
Step 1: Find cos(x)
Given that cos(x) = -8/17, we can use the Pythagorean identity to find sin(x):
sin(x) = ±√(1 - cos^2(x))
sin(x) = ±√(1 - (-8/17)^2)
sin(x) = ±√(1 - 64/289)
sin(x) = ±√((289 - 64)/289)
sin(x) = ±√(225/289)
sin(x) = ±√(225)/√(289)
sin(x) = ±15/17
Step 2: Find sin(x/2)
Using the half-angle identity for sin(x/2):
sin(x/2) = ±√((1 - cos(x))/2)
sin(x/2) = ±√((1 - (-8/17))/2)
sin(x/2) = ±√((1 + 8/17)/2)
sin(x/2) = ±√(25/17)/√2
sin(x/2) = ±(5)/(√(17) * √(2))
sin(x/2) = ±(5√2)/(√(17))
Step 3: Find cos(x/2)
Using the half-angle identity for cos(x/2):
cos(x/2) = ±√((1 + cos(x))/2)
cos(x/2) = ±√((1 + (-8/17))/2)
cos(x/2) = ±√((17/17 - 8/17)/2)
cos(x/2) = ±√((9/17)/2)
cos(x/2) = ±√(9/34)
cos(x/2) = ±(3)/(√(34) * √(2))
cos(x/2) = ±(3√2)/(√(34))
Step 4: Find tan(x/2)
Using the formula tan(x/2) = sin(x/2) / cos(x/2):
tan(x/2) = ((±(5√2)/(√(17))) / (±(3√2)/(√(34))))
tan(x/2) = ((±(5√2)/(√(17))) * (√(34))/(±(3√2)))
tan(x/2) = (±5√2 * √34)/(±3√2 * √17)
tan(x/2) = (±5 * (√2 * √34))/(±3 * (√2 * √17))
tan(x/2) = (±5 * √(2*34))/(±3 * √(2*17))
tan(x/2) = (±5√(68))/(±3√(34))
tan(x/2) = (±5√(68))/(±3√(34))
Please note that the sign (±) can be positive or negative, depending on the quadrant of x.
So, the final answers are:
sin(x/2) = (±5√2)/(√(17))
cos(x/2) = (±3√2)/(√(34))
tan(x/2) = (±5√(68))/(±3√(34))