How do you use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle
157
∘
30
'
?
If your question mean:
Determine the exact values of sine, cosine, and tangent of the angle 157° 30´ then:
157° 30´ = 180° - 22° 30´ = 180° - 45° / 2
sin ( 180° - θ ) = sin θ
sin 157° 30´ = sin ( 180° - 22° 30´ ) = sin ( 180° - 45° / 2 ) = sin 45° / 2
sin 157° 30´ = sin 45° / 2
cos ( 180° - θ ) = - cos θ
cos 157° 30´ = - cos ( 180° - 22° 30´ ) = - cos ( 180° - 45° / 2 ) = - cos 45° / 2
cos 157° 30´ = - cos 45° / 2
tan ( 180° - θ ) = - tan θ
tan 157° 30´ = tan ( 180° - 22° 30´ ) = tan ( 180° - 45° / 2 ) = - tan 45° / 2
tan 157° 30´ = - tan 45° / 2
Using the half angle identitys:
sin ( x / 2 ) = ± √ [ ( 1 - cos x ) / 2 ]
sin ( 45° / 2 ) = ± √ [ ( 1 - cos 45° ) / 2 ]
In the first quadrant all trigonometric functions are positive, so:
sin ( 45° / 2 ) = √ [ ( 1 - cos 45° ) / 2 ] =
√ [ ( 1 - √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 - √ 2 / 2 ) / 2 ] =
√ [ ( 2 - √ 2 ) / 2 / 2 ] = √ [ ( 2 - √ 2 ) / 4 ] =
√ ( 2 - √ 2 ) / √ 4 = √ ( 2 - √ 2 ) / 2
cos ( x / 2 ) = ± √ [ ( 1 + cos x ) / 2 ]
cos ( 45° / 2 ) = ± √ [ ( 1 + cos 45° ) / 2 ]
In the first quadrant all trigonometric functions are positive, so:
cos ( 45° / 2 ) = √ [ ( 1 + cos 45° ) / 2 ] =
√ [ ( 1 + √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 + √ 2 / 2 ) / 2 ] =
√ [ ( 2 + √ 2 ) / 2 / 2 ] = √ [ ( 2 + √ 2 ) / 4 ] =
√ ( 2 + √ 2 ) / √ 4 = √ ( 2 + √ 2 ) / 2
tan ( x / 2 ) = ± √ [ ( 1 - cos x ) / ( 1 + cos x ) ]
tan ( 45° / 2 ) = ± √ [ ( 1 - cos 45° ) / ( 1 + cos 45° ) ]
In the first quadrant all trigonometric functions are positive, so:
tan ( 45° / 2 ) = √ [ ( 1 - cos 45° ) / ( 1 + cos 45° ) ]
tan ( 45° / 2 ) = √ [ ( 1 - cos 45° ) / ( 1 + cos 45° ) ] =
√ [ ( 1 - √ 2 / 2 ) / ( 1 + √ 2 / 2 ) ] = √ [ ( 2 / 2 - √ 2 / 2 ) / ( 2 / 2 + √ 2 / 2 ) ] =
√ [ ( 2 - √ 2 ) / 2 / ( 2 + √ 2 ) / 2 ] = √ [ ( 2 - √ 2 ) / ( 2 + √ 2 ) ] =
√ [ ( √ 2 ∙ √ 2 - √ 2 ) / ( √ 2 ∙ √ 2 + √ 2 ) ] =
√ [ √ 2 ∙ ( √ 2 - 1 ) / ( √ 2 ∙ ( √ 2 + 1 ) ] =
√ [ ( √ 2 - 1 ) / ( √ 2 + √1 ) ] =
√ [ ( √ 2 - 1 ) ∙ ( √ 2 - 1 ) / ( √ 2 + 1 ) ∙ ( √ 2 - 1 ) ] =
√ ( √ 2 - 1 )² / √ [ ( √ 2 )² - 1² ) ] =
( √ 2 - 1 ) / √ ( 2 - 1 ) =
( √ 2 - 1 ) / √ 1 = ( √ 2 - 1 ) / 1 = √ 2 - 1
So:
sin 157° 30´ = sin 45° / 2
sin 157° 30´ = √ ( 2 - √ 2 ) / 2
cos 157° 30´ = - cos 45° / 2
cos 157° 30´ = - √ ( 2 + √ 2 ) / 2
tan 157° 30´ = - tan 45° / 2
tan 157° 30´ = - ( √ 2 - 1 ) = - √ 2 + 1
tan 157° 30´ = 1 - √ 2
157.5° = (1/2)315°
sin315 = -sin45 = -√2/2 and cos315 = cos45 = √2/2
using cos 2A = 1 - 2sin^2 A or 2cos^2 A - 1
cos 315 = 1 - 2sin^2 157.5°
√2/2 = 1 - 2sin^2 157.5°
sin^2 157.5° = 1 - √2/2 = (2 - √2)/4
Sin 157.5° = √ (2 - √2)/2
cos315 = 2cos^2 157.5 - 1
cos^2 157.5 = (√2/2 + 1)/2 = (√2 + 2)/4
cos 157.5 = - √(√2 + 2) / 2 , because 157.5 is in quad II
tan 157.5° = Sin 157.5°/cos 157.5°
= [√ (2 - √2)/2]/[- √(√2 + 2) / 2] = - √[ (2 - √2)/(√2 + 2) ]
which after rationalizing reduces to Bosnian's answer of 1 - √ 2
Well, "Clowns don't do half angles, they're all or nothing! But I'll see what I can do!
To determine the exact values of sine, cosine, and tangent of the angle 157°30', we'll need to use the half angle formulas.
First, let's split the angle 157°30' into two halves.
157°30' ÷ 2 = 78°45'
Now, let's find the exact values using the half angle formulas:
1. Sine (θ/2) = ±√[(1 - cos θ) / 2]
Sine (78°45') = ±√[(1 - cos 157°30') / 2]
Hey, wait a second! We can't determine the exact value of cosine for 157°30' just by using the half angle formulas. Looks like it's a bit of a tricky question. Sorry about that!
But hey, who needs exact values when we have approximate values and a good sense of humor, right?"
To use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle 157∘30', we first need to express the angle in radians.
To convert the angle from degrees to radians, we use the formula:
radians = degrees * π / 180
So, 157∘30' in radians would be:
radians = (157 + 30/60) * π / 180
radians = 157.5 * π / 180
Now, let's use the half angle formulas:
1. Sine formula for half angle:
sin(θ/2) = ±√((1 - cosθ) / 2)
2. Cosine formula for half angle:
cos(θ/2) = ±√((1 + cosθ) / 2)
3. Tangent formula for half angle:
tan(θ/2) = sinθ / (1 + cosθ)
In our case, the angle θ is 157.5 * π / 180.
To determine the exact values, we need to substitute the value of θ in the above formulas and calculate the result.
To use the half angle formulas to determine the exact values of sine, cosine, and tangent of an angle, follow these steps:
1. Convert the angle to radians:
157° 30' = 157 + 30/60 = 157.5 degrees
To convert degrees to radians, multiply by π/180:
157.5 * π/180 = 2.7475 radians (approximately)
2. Use the half angle formulas:
- Sine of half angle:
sin(x/2) = ± √((1 - cos(x)) / 2)
- Cosine of half angle:
cos(x/2) = ± √((1 + cos(x)) / 2)
- Tangent of half angle:
tan(x/2) = sin(x) / (1 + cos(x))
3. Substitute the angle in radians into the formulas:
- Sine of half angle:
sin(2.7475/2) = ± √((1 - cos(2.7475)) / 2)
- Cosine of half angle:
cos(2.7475/2) = ± √((1 + cos(2.7475)) / 2)
- Tangent of half angle:
tan(2.7475/2) = sin(2.7475) / (1 + cos(2.7475))
4. Calculate the exact values using a calculator:
Evaluate the square roots, cosine, sine, and tangent functions using a calculator or a software that supports trigonometric functions.
Note: The ± sign indicates that there are two possible solutions, one positive and one negative. The choice of sign depends on the quadrant in which the angle lies.