1) use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.
67 degrees and 30'
Please explain!! I do not know how to start.
Notice that 67 degrees, 30' is half of 135 degrees,
and 135 = 180-45
We also know that
cos 2A = 2 cos^2 A - 1
giving us
cos 135 = 2cos^2 67.5 - 1
So let's find cos 135
135 is in quadrant II, so
cos 135 = -sin 45 = -1/√2
back to cos 135 = 2cos^2 67.5 - 1
-1/√2 = 2cos^2 67.5 - 1
1 - 1/√2 = 2cos^2 67.5
(√2-1)/(2√2) = cos^2 67.5
cos 67.5 = √[(√2-1)/(2√2)]
(my calculator confirmed my answer to be correct
Using the other version of
cos 2A = 1 - 2sin^2 A -1 and following the above steps will give you the sin 67.5
you then find tan 67.5 by sin67.5/cos67.5 using the expressions.
Enjoy the algebra, lol
To use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of an angle, we first need to identify the corresponding half-angle formulas.
The half-angle formulas for sine, cosine, and tangent are derived from the double-angle formulas and they allow us to find the exact trigonometric values for angles that are half of a given angle.
Let's start by finding the half-angle for the given angle, which is 67 degrees and 30 minutes.
To convert the minutes to degrees, we divide by 60, so 30 minutes can be expressed as 30/60 = 0.5 degrees. So, the given angle is 67.5 degrees.
To find the half-angle, we divide the given angle by 2: 67.5 degrees / 2 = 33.75 degrees.
Now that we have the half-angle, we can apply the half-angle formulas.
Sine half-angle formula: sin(x/2) = ±√[(1 - cos(x)) / 2]
Cosine half-angle formula: cos(x/2) = ±√[(1 + cos(x)) / 2]
Tangent half-angle formula: tan(x/2) = ±√[(1 - cos(x)) / (1 + cos(x))]
Let's use these formulas to determine the exact values.
1. Sine:
Using the sine half-angle formula, we have:
sin(33.75 degrees/2) = ±√[(1 - cos(33.75 degrees)) / 2]
To find cos(33.75 degrees), we can use a calculator or reference table. The value of cos(33.75 degrees) is approximately 0.923.
Plugging this value into the formula, we get:
sin(33.75 degrees/2) = ±√[(1 - 0.923) / 2]
sin(33.75 degrees/2) = ±√[0.077 / 2]
sin(33.75 degrees/2) = ±√0.0385
sin(33.75 degrees/2) ≈ ±0.196
So, the exact value of the sine of the angle 67 degrees and 30 minutes is approximately ±0.196.
2. Cosine:
Using the cosine half-angle formula, we have:
cos(33.75 degrees/2) = ±√[(1 + cos(33.75 degrees)) / 2]
Similarly, we find cos(33.75 degrees) ≈ 0.923.
Plugging this value into the formula, we get:
cos(33.75 degrees/2) = ±√[(1 + 0.923) / 2]
cos(33.75 degrees/2) = ±√[1.923 / 2]
cos(33.75 degrees/2) = ±√0.9615
cos(33.75 degrees/2) ≈ ±0.981
So, the exact value of the cosine of the angle 67 degrees and 30 minutes is approximately ±0.981.
3. Tangent:
Using the tangent half-angle formula, we have:
tan(33.75 degrees/2) = ±√[(1 - cos(33.75 degrees)) / (1 + cos(33.75 degrees))]
Plugging in the value for cos(33.75 degrees) ≈ 0.923, we get:
tan(33.75 degrees/2) = ±√[(1 - 0.923) / (1 + 0.923)]
tan(33.75 degrees/2) = ±√[0.077 / 1.923]
tan(33.75 degrees/2) = ±√0.040
tan(33.75 degrees/2) ≈ ±0.200
So, the exact value of the tangent of the angle 67 degrees and 30 minutes is approximately ±0.200.
To use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of an angle, follow these steps:
Step 1: Convert the angle from degrees and minutes into decimal degrees.
In this case, the angle is 67 degrees and 30 minutes. To convert minutes to decimal form, you divide the number of minutes by 60. So, 30 minutes divided by 60 is equal to 0.5 degrees. Therefore, the angle in decimal form is 67.5 degrees.
Step 2: Use the half-angle formulas to find the values of sin, cos, and tan for half the angle.
The half-angle formulas are:
- Sin(θ/2) = ± sqrt((1 - cosθ) / 2)
- Cos(θ/2) = ± sqrt((1 + cosθ) / 2)
- Tan(θ/2) = sinθ / (1 + cosθ)
Step 3: Apply the formulas for half the angle using the converted decimal angle.
For the angle of 67.5 degrees, you need to find the values of sin, cos, and tan for half of this angle. Therefore, divide 67.5 by 2 to find half the angle, which is 33.75 degrees.
Step 4: Calculate sin(33.75/2):
sin(33.75/2) = ± sqrt((1 - cos33.75) / 2)
To calculate the exact value of cos33.75, you may use a calculator or a trigonometric table. However, for the purpose of this explanation, let's assume a calculator is used and we obtain cos33.75 ≈ 0.934.
Now substitute this value into the formula:
sin(33.75/2) = ± sqrt((1 - 0.934) / 2)
sin(33.75/2) = ± sqrt(0.033 / 2)
sin(33.75/2) = ± sqrt(0.0165)
Taking the square root of 0.0165 is approximately equal to 0.128.
Thus, sin(33.75/2) = ± 0.128.
Step 5: Calculate cos(33.75/2):
cos(33.75/2) = ± sqrt((1 + cos33.75) / 2)
Again, assuming that cos33.75 ≈ 0.934:
cos(33.75/2) = ± sqrt((1 + 0.934) / 2)
cos(33.75/2) = ± sqrt(1.934 / 2)
cos(33.75/2) = ± sqrt(0.967)
Taking the square root of 0.967 is approximately equal to 0.983.
Thus, cos(33.75/2) = ± 0.983.
Step 6: Calculate tan(33.75/2):
tan(33.75/2) = sin(33.75) / (1 + cos(33.75))
Using the values obtained earlier:
- sin(33.75/2) = ± 0.128
- cos(33.75/2) = ± 0.983
Substitute the values into the formula:
tan(33.75/2) = ± 0.128 / (1 + 0.983)
tan(33.75/2) = ± 0.128 / 1.983
tan(33.75/2) ≈ ± 0.0646
Thus, tan(33.75/2) ≈ ± 0.0646.
Note: The ± symbol in front of the values indicates that both positive and negative values are possible. The sign depends on the quadrant in which the angle lies.