b) convert each angle to radians.
i) 150°
ii) −240°
iii) 485°
5. in a circle with radius 5 cm, an arc of length 6 cm subtends a central angle. what is the measure of this angle in radians, and to the nearest degree?
6. a race car is travelling around a circular track at an average speed of 120 km/h. the track has a diameter of 1 km. visualize a line segment joining the race car to the centre of the track. through what angle, in radians, will the segment have rotated in 10 s?
7. determine the value of each trigonometric ratio. use exact values where possible; otherwise write the value to the nearest thousandth

Question

b) convert each angle to radians.
 i) 150°
 ii) −240°
 iii) 485°
 5. in a circle with radius 5 cm, an arc of length 6 cm subtends a central angle. what is the measure of this angle in radians, and to the nearest degree?
 6. a race car is travelling around a circular track at an average speed of 120 km/h. the track has a diameter of 1 km. visualize a line segment joining the race car to the centre of the track. through what angle, in radians, will the segment have rotated in 10 s?
 7. determine the value of each trigonometric ratio. use exact values where possible; otherwise write the value to the nearest thousandth

Accepted Answer

### b) Converting angles to radians
#### i) $150^\circ$
# Explanation:
## Step1: Recall the conversion formula
To convert degrees to radians, we use the formula: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
## Step2: Apply the formula
For $150^\circ$, we have: $150 \times \frac{\pi}{180}$
Simplify the fraction: $\frac{150\pi}{180} = \frac{5\pi}{6}$
# Answer: $\frac{5\pi}{6}$ radians

#### ii) $-240^\circ$
# Explanation:
## Step1: Recall the conversion formula
$\text{radians} = \text{degrees} \times \frac{\pi}{180}$
## Step2: Apply the formula
For $-240^\circ$, we have: $-240 \times \frac{\pi}{180}$
Simplify the fraction: $\frac{-240\pi}{180} = -\frac{4\pi}{3}$
# Answer: $-\frac{4\pi}{3}$ radians

#### iii) $485^\circ$
# Explanation:
## Step1: Recall the conversion formula
$\text{radians} = \text{degrees} \times \frac{\pi}{180}$
## Step2: Apply the formula
For $485^\circ$, we have: $485 \times \frac{\pi}{180}$
Simplify the fraction: $\frac{485\pi}{180} = \frac{97\pi}{36}$ (or approximately $8.49$ radians)
# Answer: $\frac{97\pi}{36}$ radians (or $\approx 8.49$ radians)

### 5. Arc length and central angle
# Explanation:
## Step1: Recall the arc length formula
The formula for arc length $s$ is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.
## Step2: Solve for $\theta$
We know $s = 6$ cm and $r = 5$ cm. Rearranging the formula: $\theta = \frac{s}{r}$
Substitute the values: $\theta = \frac{6}{5} = 1.2$ radians
## Step3: Convert radians to degrees
To convert radians to degrees, use the formula: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$
So, $\theta_{\text{degrees}} = 1.2 \times \frac{180}{\pi} \approx \frac{216}{\pi} \approx 68.75^\circ$ (to the nearest degree)
# Answer: The central angle is $1.2$ radians, or approximately $69^\circ$ (to the nearest degree)

### 6. Angle rotated by the race car
# Explanation:
## Step1: Find the circumference of the track
The diameter $d = 1$ km, so the radius $r = \frac{d}{2} = 0.5$ km. The circumference $C = 2\pi r = 2\pi(0.5) = \pi$ km.
## Step2: Find the distance traveled in 10 seconds
The speed is $120$ km/h. Convert 10 seconds to hours: $10$ seconds $= \frac{10}{3600} = \frac{1}{360}$ hours.
Distance $d = \text{speed} \times \text{time} = 120 \times \frac{1}{360} = \frac{1}{3}$ km.
## Step3: Find the angle in radians
The angle $\theta$ (in radians) is given by the ratio of the distance traveled to the circumference, multiplied by $2\pi$ (since a full circle is $2\pi$ radians). Or, using the arc length formula $s = r\theta$, where $s$ is the distance traveled, $r$ is the radius.
Using $s = r\theta$: $\theta = \frac{s}{r}$
We have $s = \frac{1}{3}$ km and $r = 0.5$ km. So, $\theta = \frac{\frac{1}{3}}{0.5} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}$ radians.
# Answer: The angle rotated is $\frac{2}{3}$ radians (or approximately $1.047$ radians)

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QUESTION IMAGE

Question

Explanation:

b) Converting angles to radians

i) \(150^\circ\)

Step1: Recall the conversion formula

Step2: Apply the formula

Step1: Recall the conversion formula

Step2: Apply the formula

Step1: Recall the conversion formula

Step2: Apply the formula

Answer:

ii) \(-240^\circ\)