Haruto simplified the value below.
[tex]$\csc \left(\frac{19 \pi}{4}\right)=\sqrt{2}$[/tex]
Which statement explains whether Haruto is correct?
A. Haruto is correct because the angle is coterminal with [tex]$\frac{3 \pi}{4}$[/tex] and the reference angle is [tex]$\frac{\pi}{4}$[/tex].
B. Haruto is correct because the angle is coterminal with [tex]$\frac{\pi}{4}$[/tex], which is also the reference angle.
C. Haruto is not correct because the angle is coterminal with [tex]$\frac{3 \pi}{4}$[/tex], and [tex]$\csc \left(\frac{3 \pi}{4}\right)=\frac{-1}{\sqrt{2}}=\frac{-\sqrt{2}}{2}$[/tex].
D. Haruto is not correct because the angle is coterminal with [tex]$\frac{\pi}{4}$[/tex], and [tex]$\csc \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$[/tex].
Answer (2)
Find the coterminal angle of 4 19 π , which is 4 3 π .
Determine the reference angle for 4 3 π , which is 4 π .
Calculate sin ( 4 3 π ) = 2 2 .
Calculate csc ( 4 3 π ) = s i n ( 4 3 π ) 1 = 2 , so the answer is Haruto is correct because the angle is coterminal with 4 3 π and the reference angle is 4 π .
Explanation
Problem Analysis We are given that Haruto simplified csc ( 4 19 π ) to 2 . We need to determine if Haruto is correct and choose the correct explanation.
Finding Coterminal Angle First, let's find the coterminal angle of 4 19 π . To do this, we subtract multiples of 2 π = 4 8 π until we get an angle between 0 and 2 π .
First Subtraction 4 19 π − 4 8 π = 4 11 π
Second Subtraction 4 11 π − 4 8 π = 4 3 π
Coterminal Angle So, 4 19 π is coterminal with 4 3 π .
Calculating Cosecant Now, let's calculate csc ( 4 3 π ) . Since csc ( x ) = s i n ( x ) 1 , we need to find sin ( 4 3 π ) .
Finding Reference Angle The reference angle for 4 3 π is π − 4 3 π = 4 π .
Sine of Reference Angle sin ( 4 3 π ) = sin ( 4 π ) = 2 2
Cosecant Calculation Therefore, csc ( 4 3 π ) = s i n ( 4 3 π ) 1 = 2 2 1 = 2 2 = 2
Conclusion Since csc ( 4 19 π ) = csc ( 4 3 π ) = 2 , Haruto is correct. The angle is coterminal with 4 3 π and the reference angle is 4 π .
Examples
Cosecant, sine, and coterminal angles are useful in fields like electrical engineering when analyzing alternating current (AC) circuits. For example, when modeling the voltage or current in an AC circuit, we often use sinusoidal functions. Understanding coterminal angles helps simplify calculations when dealing with periodic signals that repeat every 2 π radians. Also, in navigation, these concepts are used to determine the direction and position of objects, especially when dealing with angles on a circular path.
Haruto's claim that csc ( 4 19 π ) = 2 is correct. The angle is coterminal with 4 3 π and its reference angle is 4 π . Therefore, the correct choice is A.
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