Use the half-angle formulas to find the exact values. Cosine 165 degrees, Sine 157 degrees 30' and tan pi/8

To find the exact values using the half-angle formulas, we can use the formulas:

1. Cosine half-angle formula:
cos(x/2) = ±√[(1 + cos(x))/2]

2. Sine half-angle formula:
sin(x/2) = ±√[(1 - cos(x))/2]

3. Tangent half-angle formula:
tan(x/2) = ±√[(1 - cos(x))/(1 + cos(x))]

Let's solve for each value step-by-step:

1. Cosine 165 degrees:
Angle 165 degrees can be represented as the sum of 180 degrees and -15 degrees.

cos(165) = cos(180 - 15)

Now, we can use the cosine half-angle formula:
cos(x/2) = ±√[(1 + cos(x))/2]

cos(165/2) = ±√[(1 + cos(165))/2]

Since cos(180 - 15) = cos(15), we can substitute cos(165) with cos(15) in the formula:
cos(165/2) = ±√[(1 + cos(15))/2]

Note: We can use the positive square root in this case since 165 degrees is in the second quadrant where the cosine value is negative.

Therefore, cos(165/2) = -√[(1 + cos(15))/2]

2. Sine 157 degrees 30':
To convert minutes to degrees, we divide by 60:
30' / 60 = 0.5 degrees

Angle 157 degrees 30' can be represented as the sum of 157 degrees and 0.5 degrees.

sin(157' 30") = sin(157 + 0.5)

Now, we can use the sine half-angle formula:
sin(x/2) = ±√[(1 - cos(x))/2]

sin(157.5/2) = ±√[(1 - cos(157.5))/2]

Note: Since 157.5 degrees is in the second quadrant where the sine value is positive, we can use the positive square root.

Therefore, sin(157.5/2) = √[(1 - cos(157.5))/2]

3. Tangent pi/8:
The angle pi/8 can be converted to degrees using the fact that π radians is equal to 180 degrees:
pi radians = 180 degrees

Therefore, pi/8 = (180/π)/8 = 22.5 degrees

Now, we can use the tangent half-angle formula:
tan(x/2) = ±√[(1 - cos(x))/(1 + cos(x))]

tan(22.5/2) = ±√[(1 - cos(22.5))/(1 + cos(22.5))]

Since 22.5 degrees is in the first quadrant where the tangent value is positive, we can use the positive square root.

Therefore, tan(22.5/2) = √[(1 - cos(22.5))/(1 + cos(22.5))]

So, the exact values using the half-angle formulas are:

1. cos(165) = -√[(1 + cos(15))/2]
2. sin(157' 30") = √[(1 - cos(157.5))/2]
3. tan(pi/8) = √[(1 - cos(22.5))/(1 + cos(22.5))]

To find the exact values using the half-angle formulas, we need to first determine the reference angle for each given angle.

1. For cosine 165 degrees:
The reference angle can be found by subtracting 180 degrees from the given angle since cosine is negative in the second quadrant.
Reference angle = 165 degrees - 180 degrees = -15 degrees

The half-angle formula for cosine is:
cos(x/2) = ±√((1 + cos(x))/2)

Substitute the reference angle into the formula:
cos(-15/2) = ±√((1 + cos(-15))/2)

Now, we need to determine the cosine of -15 degrees. To do this, we can use the fact that cosine is an even function, meaning cosine(-θ) = cosine(θ). Therefore, cos(-15) = cos(15).

cos(-15/2) = ±√((1 + cos(15))/2)

Lastly, we need to determine the exact value of cos(15). Since 15 degrees is not a commonly known angle, we can use a calculator to find its approximate value and then simplify it.

If you use a calculator, you will find that cos(15) ≈ 0.9659.

So, the final result for cos(165 degrees) using the half-angle formula is:
cos(165 degrees) = ±√((1 + 0.9659)/2)
= ±√((1.9659)/2)
≈ ±0.692

Therefore, cos(165 degrees) ≈ ±0.692.

2. For sine 157 degrees 30':
The reference angle can be found by subtracting 180 degrees from the given angle since sine is negative in the third quadrant.
Reference angle = 157 degrees 30' - 180 degrees = -22 degrees 30'

The half-angle formula for sine is:
sin(x/2) = ±√((1 - cos(x))/2)

Substitute the reference angle into the formula:
sin(-22/2) = ±√((1 - cos(-22))/2)

Like before, we need to determine the cosine of -22 degrees. Using the fact that cosine is an even function, cos(-22) = cos(22).

sin(-22/2) = ±√((1 - cos(22))/2)

Find the approximate value of cos(22) using a calculator and simplify:
cos(22) ≈ 0.9272

so, the final result for sin(157 degrees 30') using the half-angle formula is:
sin(157 degrees 30') = ±√((1 - 0.9272)/2)
= ±√((0.0728)/2)
≈ ±0.270

Therefore, sin(157 degrees 30') ≈ ±0.270.

3. For tan(pi/8):
The half-angle formula for tangent is:
tan(x/2) = ±√((1 - cos(x))/(1 + cos(x)))

Substitute the given angle into the formula:
tan(pi/8) = ±√((1 - cos(pi/4))/(1 + cos(pi/4)))

Simplify the cosine values using the known values for pi/4 (45 degrees - a commonly known angle):
cos(pi/4) = cos(45 degrees) = 1/√2

tan(pi/8) = ±√((1 - 1/√2)/(1 + 1/√2))

Rationalize the denominator:
tan(pi/8) = ±√((√2 - 1)/√2 * √2/√2)/(√2 + 1)/(√2/√2))
= ±√((2 - √2)/(2 + √2))

Multiply numerator and denominator by the conjugate of the denominator to simplify further:
tan(pi/8) = ±√((2 - √2)^2/(2^2 - (√2)^2))
= ±√((4 - 2√2 + 2)/(4 - 2))
= ±√((6 - 2√2)/2)
= ±√(3 - √2)

Therefore, tan(pi/8) = ±√(3 - √2).