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Zorluk: OrtaRadians and Degrees

A circular dial on a vintage radio is rotated by 5π8\frac{5\pi}{8} radians to tune to a specific station. If the dial is then rotated by an additional 45∘45^\circ in the same direction, what is the total angle of rotation, in radians, of the dial?

  1. A
    3π4\frac{3\pi}{4}
  2. 7π8\frac{7\pi}{8}Cevap
  3. C
    9π8\frac{9\pi}{8}
  4. D
    π2\frac{\pi}{2}

Cevap

The correct answer is 7π8\frac{7\pi}{8} radians.
To find the total angle of rotation in radians, the rotation of 45∘45^\circ must first be converted to radians by multiplying by π180∘\frac{\pi}{180^\circ}, which yields π4\frac{\pi}{4} radians. Expressed with a common denominator of 88, this is equivalent to 2π8\frac{2\pi}{8} radians. Adding this to the initial rotation of 5π8\frac{5\pi}{8} radians gives a total rotation of 5π8+2π8=7π8\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8} radians.

Adım Adım Çözüm

1
Convert the additional rotation angle of 45∘45^\circ into radians.
Since 180∘=π180^\circ = \pi radians, the conversion is 45∘×π180∘=π445^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4} radians.
To find the total rotation angle in radians, both individual angles must be in the same unit of measure.
2
Add the first rotation angle of 5π8\frac{5\pi}{8} radians to the converted second rotation angle of π4\frac{\pi}{4} radians.
First, express π4\frac{\pi}{4} with a common denominator of 88: π4=2π8\frac{\pi}{4} = \frac{2\pi}{8}. Then, add the two fractions: 5π8+2π8=7π8\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8} radians.
The total angle of rotation is the sum of the two sequential rotations in the same direction.

Anahtar Kavram

To find the sum of angles given in different units, convert the angle measured in degrees to radians using the conversion factor π radians180∘\frac{\pi \text{ radians}}{180^\circ}, and then find the sum of the two radian values using a common denominator.
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Radians and Degrees

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