Question

exercices 5

1. quelle est la mesure, en radians des angles suivants ?

a) 23,456°
b) -42°4123
c) 270°1342

1. quelle est la mesure, en degré décimal, des angles suivants ?

a) 4,28 rad
b) 0,25 rad
c) -80°305

1. un côté de votre terrain décrit un arc de 42,3 m de longueur. si le rayon de courbure est de 55,9 m, quelle est la grandeur de langle central, en degré - minute - seconde, sous - tendu par cet arc de cercle ?

Explanation:

1. Question 1: Convert degrees to radians

• Recall the conversion formula: To convert degrees to radians, use the formula $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$.
• a) For $\theta = 23.456^{\circ}$
• # Explanation:

Step1: Apply the conversion formula

$\theta_{rad}=23.456\times\frac{\pi}{180}$
$\theta_{rad}\approx23.456\times\frac{3.14159}{180}\approx0.4094$ rad

• b) For $\theta=- 42^{\circ}41'23''$
• First, convert minutes and seconds to degrees.
• 1 minute $=\frac{1}{60}$ degrees and 1 second $=\frac{1}{3600}$ degrees.
• $41' = 41\times\frac{1}{60}\approx0.6833^{\circ}$ and $23''=23\times\frac{1}{3600}\approx0.0064^{\circ}$
• So, $\theta=-42^{\circ}+ 0.6833^{\circ}+0.0064^{\circ}=-42.6897^{\circ}$
• # Explanation:

Step1: Apply the conversion formula

$\theta_{rad}=-42.6897\times\frac{\pi}{180}$
$\theta_{rad}\approx - 42.6897\times\frac{3.14159}{180}\approx - 0.7450$ rad

• c) For $\theta = 270^{\circ}13'42''$
• Convert minutes and seconds to degrees: $13'=13\times\frac{1}{60}\approx0.2167^{\circ}$ and $42'' = 42\times\frac{1}{3600}\approx0.0117^{\circ}$
• $\theta=270^{\circ}+0.2167^{\circ}+0.0117^{\circ}=270.2284^{\circ}$
• # Explanation:

Step1: Apply the conversion formula

$\theta_{rad}=270.2284\times\frac{\pi}{180}$
$\theta_{rad}\approx270.2284\times\frac{3.14159}{180}\approx4.7156$ rad

1. Question 2: Convert radians to decimal - degrees

• Recall the conversion formula: $\theta_{deg}=\theta_{rad}\times\frac{180}{\pi}$
• a) For $\theta = 4.28$ rad
• # Explanation:

Step1: Apply the conversion formula

$\theta_{deg}=4.28\times\frac{180}{\pi}$
$\theta_{deg}\approx4.28\times\frac{180}{3.14159}\approx245.22^{\circ}$

• b) For $\theta = 0.25$ rad
• # Explanation:

Step1: Apply the conversion formula

$\theta_{deg}=0.25\times\frac{180}{\pi}$
$\theta_{deg}\approx0.25\times\frac{180}{3.14159}\approx14.32^{\circ}$

• c) For $\theta=-80^{\circ}30'5''$
• First, convert the given angle to decimal - degrees: $30' = 30\times\frac{1}{60}=0.5^{\circ}$ and $5''=5\times\frac{1}{3600}\approx0.0014^{\circ}$
• $\theta=-80^{\circ}-0.5^{\circ}-0.0014^{\circ}=-80.5014^{\circ}$
• This part seems to be a wrong - input as it is given in degrees already while the question asks to convert radians to degrees. If we assume it was a mis - type and we need to convert it to radians:
• $\theta_{rad}=-80.5014\times\frac{\pi}{180}\approx - 1.4050$ rad

Answer:

1.

• a) Approximately $0.4094$ rad
• b) Approximately $-0.7450$ rad
• c) Approximately $4.7156$ rad

2.

• a) Approximately $245.22^{\circ}$
• b) Approximately $14.32^{\circ}$
• c) There is a possible mis - input. If converting the given degrees to radians, approximately $-1.4050$ rad.