The size of angle θ expressed in radians equals s (the arc length subtendend by the angle) divided by r (the radius of the circle). |
c^{2} = a^{2} + b^{2}A nice animated proof of Pythagoras’ theorem can be found at http://www.math.ubc.ca/~morey/java/pyth/
c^{2} = 3^{2} + 3.5^{2} = 21.25Taking the square root of both sides gives the length of the straight line as
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C = 180° − 32.5° − 49.7° = 97.8°Now use the sine law to get sides b and c:
A = 25.3°, c = 152 and a = 95.0
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V = 2.5 ∠ 53°∠ is called the angle symbol. The number in front of the angle symbol is the magnitude or length of the vector. The number after the angle symbol is the direction of the vector, expressed as an angle measured counterclockwise from the positive part of the real axis.
V = (1.5, 2)
A + B = ( 2, 4 ) + ( 3, 1 ) = ( 5, 5 )The picture shows that this works because:
A + B = ( 2, 4 ) + ( 3, 1 )
= ( 2, 0 ) + ( 0, 4 ) + ( 3, 0 ) + ( 0, 1 )
= ( 2, 0 ) + ( 3, 0 ) + ( 0, 4 ) + ( 0, 1 )
= ( 5, 0 ) + ( 0, 5 )
= ( 5, 5 )
x = r cos(θ) and y = r sin(θ).
8 ∠ 30° + 10 ∠ −60° = 12.8 ∠ 21.3°
1 ∠ 50° + 2 ∠ 50° = 3 ∠ 50°
(a) What are the tensions T_{1} and T_{2} in the ropes?Solution: The forces T_{1} , T_{2} and W must be in balance:
(b) If the ropes have a tensile strength of 1000 Newtons, what is the maximum weight that can be supported?
F = W sin(θ) = (1250 N) sin(13°) = 281 NThus the minimum force is 281 Newtons.
A = π r^{2}The area of a sector is a fraction s/C of the area of a circle, where C is the circumference. Thus:
s = r θ = (3960 mi)(0.7610 rads) = 3010 mi,to 3 sig figs. Notice that the units of s are the same as those of r (miles), since the units of θ (radians) are actually unitless.
s = r θ = (11.25 mm)(5.236 rads) = 58.9 mm.Notice that the units of s are the same as those of r, namely millimeters.
θ = ω t = (48 π rads / sec)(0.25 sec) = 12 π rads.(Note that 48 π rads / sec = 24 cycles / sec and that 12 π rads = 6 cycles)
, and ,where r is the length of a vector and x and y are its x and y components. Let r = 1. Then sin(θ) = y and cos(θ) = x. In other words sin(θ) is just the y component and cos(θ) is the x component of a vector of length 1.
y = A sin(ω t + φ)where y represents the quantity of interest and t represents time. Because y is proportional to the sin function, it has the characteristic sin wave shape. By changing A, ω and φ we can change the height or width of the wave or shift it left or right to suit the application. Let’s look at each of these parameters, A, ω and φ, in turn:
ω = 2 π f.The units of f are cycles/sec or Hertz.
φ = 2π rads or 360°. The wave is shifted left by one complete cycle, so the wave appears unchanged.
φ = π rads or 180°. The wave is shifted left by half a cycle, so the wave appears to be turned upside-down.
φ = π/2 rads or 90°. The wave is shifted left by one quarter of a cycle, so that, for example, a sin wave becomes a cosine wave.
v = A sin(ω t + φ)
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