Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.
67° 30'
7 sin(67° 30')=
cos(67° 30')=
4 tan(67° 30')=
1. 2 x 67.5 = 135
2. tan 135 = - tan 45 = -1
3. tan 2A = [2tan A]/[1-(tan A)^2]
4. Put A = 67.5 in (3) and get
tan135 = [2tan67.5]/[1-(tan67.5)^2]
-1 = [2tan 67.5]/[1-(tan 67.5)^2]
5. Let x = tan 67.5. Then,
-1 = 2x/[1 - x^2]
x^2 - 2x - 1 = 0
By quadratic formula,
x = 1 + sqrt(2) or 1 - sqrt(2)
reject 1 - sqrt(2)since tan 67.5 > 0
6. x = tan 67.5 = 1 + sqrt(2)
4 tan 67.5 = 4 + 4 sqrt(2)
To use the half-angle formulas to determine the exact values of the trigonometric functions, we need to first convert the angle to radians.
Given angle: 67° 30'
Step 1: Convert to radians
To convert degrees to radians, use the formula:
radians = degrees * π / 180
radians = 67.5° * π / 180
radians ≈ 1.1781
Now, let's use the half-angle formulas to find the exact values of sine, cosine, and tangent.
1. Exact value of sine (sin):
The half-angle formula for sine is:
sin(x/2) = ±√[(1 - cos(x)) / 2]
Using the formula, we have:
sin(67.5°) = ±√[(1 - cos(135°)) / 2]
Since the angle is in the second quadrant (between 90° and 180°), the sine value is positive.
sin(67.5°) = √[(1 - cos(135°)) / 2]
2. Exact value of cosine (cos):
The half-angle formula for cosine is:
cos(x/2) = ±√[(1 + cos(x)) / 2]
Using the formula, we have:
cos(67.5°) = ±√[(1 + cos(135°)) / 2]
Since the angle is in the second quadrant (between 90° and 180°), the cosine value is negative.
cos(67.5°) = -√[(1 + cos(135°)) / 2]
3. Exact value of tangent (tan):
The half-angle formula for tangent is:
tan(x/2) = sin(x) / (1 + cos(x))
Using the formula, we have:
tan(67.5°) = sin(135°) / (1 + cos(135°))
Now, let's plug in the values we obtained earlier (sin(67.5°) and cos(67.5°)):
tan(67.5°) = √[(1 - cos(135°)) / 2] / (1 + cos(135°))
To simplify further, we need to find the exact values of cos(135°) and cos(67.5°).
Using the values obtained from the half-angle formulas, you can substitute them into the respective equations and calculate the exact values for sin(67° 30'), cos(67° 30'), and tan(67° 30').
To use the half-angle formulas to determine the values of sine, cosine, and tangent for the given angle, we need to convert the angle to radians and then apply the formulas.
Given angle: 67° 30'
Step 1: Convert the angle to radians
1 degree = π/180 radians
67° 30' = (67 + 30/60)° = 67.5° = (67.5 * π/180) radians
Step 2: Apply the half-angle formulas
Sine half-angle formula: sin(θ/2) = ±√((1 - cos θ)/2)
Cosine half-angle formula: cos(θ/2) = ±√((1 + cos θ)/2)
Tangent half-angle formula: tan(θ/2) = sin θ / (1 + cos θ)
Step 3: Calculate the values
sin(67° 30') = ±√((1 - cos(67.5°))/2)
cos(67° 30') = ±√((1 + cos(67.5°))/2)
tan(67° 30') = sin(67.5°) / (1 + cos(67.5°))
Now, let's calculate the values using a calculator.
sin(67° 30') ≈ ±√((1 - cos(67.5°))/2) ≈ ±0.943
cos(67° 30') ≈ ±√((1 + cos(67.5°))/2) ≈ ±0.332
tan(67° 30') ≈ sin(67.5°) / (1 + cos(67.5°)) ≈ ±2.840
So, the approximate values are:
7 sin(67° 30') ≈ ±6.601
cos(67° 30') ≈ ±0.332
4 tan(67° 30') ≈ ±11.360
Note: The ± sign indicates that there are two possible values, depending on the quadrant in which the angle falls.