a. The sine of the angle is 4.564ft.
b. The cosine of the angle is 3.640ft.
c. The tangent of the angle is 1.255.
d. The cotangent of the angle is 0.796.
How to solve for the other measurements
To solve for the other measurements of the plywood given its length of 6ft and an angle of 48 degrees, use trigonometric functions.
Let's calculate each of the requested measurements:
Use SOHCAHTOA
This means
SOH: sine theta = opposite/Hypotenuse
CAH: Cos theta = Adjacent/hypotenuse
TOA: Tan theta = opposite/adjacent
a) Sine (sin):
The sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. In this case, the opposite side is the height (h) of the plywood.
Using the formula sin(theta) = opposite/hypotenuse, we have:
sin(48 degrees) = h/6ft
Rearranging the equation to solve for h:
h = 6ft * sin(48 degrees)
h ≈ 4.564ft
Therefore, the sine of the angle is approximately 4.564ft.
b) Cosine (cos):
The cosine of an angle is the ratio of the length of the side adjacent to the angle to the hypotenuse. In this case, the adjacent side is the width (w) of the plywood.
Using the formula cos(theta) = adjacent/hypotenuse, we have:
cos(48 degrees) = w/6ft
Rearranging the equation to solve for w:
w = 6ft * cos(48 degrees)
w ≈ 3.640ft
Therefore, the cosine of the angle is approximately 3.640ft.
c) Tangent (tan):
The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, we can use the height (h) and width (w) of the plywood to calculate the tangent (t).
Using the formula tan(theta) = opposite/adjacent, we have:
tan(48 degrees) = h/w
Rearranging the equation to solve for t:
t = h/w ≈ 4.564ft / 3.640ft
t ≈ 1.255
Therefore, the tangent of the angle is approximately 1.255.
d) Cotangent (cot):
The cotangent of an angle is the reciprocal of the tangent. Use the tangent value calculated in the previous step to find the cotangent (cot).
cot(48 degrees) = 1/t ≈ 1/1.255
cot ≈ 0.796
Therefore, the cotangent of the angle is approximately 0.796.