The Principal Value (Reference Angle)
Let \( \alpha \in [0, 2\pi) \).
As seen previously, the terminal side of an angle measuring \( \alpha + 2k\pi \) radians and the terminal side of an angle measuring \( \alpha \) radians intersect the unit circle at the exact same point \( P \). This means that these angles map onto the same point on the unit circle. Here, the angle \( \alpha \) radians is called the principal value (or reference angle) of the angle \( \alpha + 2k\pi \) radians.
For Example:
If we choose \( k = 2 \), moving along a circular arc of \( 2\pi \) radians twice in the positive direction starting from point \( A \) brings us right back to point \( A \).
If we subtract \( 4\pi \) radians from \( \alpha + 4\pi \) radians, the resulting angle of \( \alpha \) radians maps onto the exact same point on the unit circle as the angle of \( \alpha + 4\pi \) radians.
Therefore, the angle \( \alpha \) radians is the principal value of the angle \( \alpha + 4\pi \) radians.
Conclusion:
If the principal value of an angle with a measure of \( A^\circ \) is \( a^\circ \) and \( a \in [0, 360) \), then:
\[
A \equiv a \pmod{360}
\]
If the principal value of an angle with a measure of \( B \) radians is \( b \) radians and \( b \in [0, 2\pi) \), then:
\[
B \equiv b \pmod{2\pi}
\]
If the principal value of an angle with a measure of \( C \) grads is \( c \) grads and \( c \in [0, 400) \), then:
\[
C \equiv c \pmod{400}
\]
In summary,
To find the principal value of an angle given in degrees, divide the measure of the angle by 360. The remainder is the principal value.
To find the principal value of an angle given in radians, subtract full multiples of \( 2\pi \) from the measure of the angle. The remainder is the principal value.
To find the principal value of an angle given in grads, divide the measure of the angle by 400. The remainder is the principal value.
Examples:
Let us find the principal values of the angles 1200°, -1200°, 19000°, and -19000°.
\(\bullet \) Principal value of the angle 1200°:
\[
\begin{array}{r|l}
1200 & 360 \\
1080 & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & 3\\
120 &
\end{array}
\]
The principal value of a 1200° angle is \( 120^\circ \).
\(\bullet \) Principal value of the angle -1200°:
\[
\begin{array}{r|l}
-1200 & 360 \\
-1440 & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & -4\\
240 &
\end{array}
\]
The principal value of a -1200° angle is \( 240^\circ \).
\(\bullet \) Principal value of the angle 19000°:
\[
\begin{array}{r|l}
19000 & 360 \\
18720 & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & 52\\
280 &
\end{array}
\]
The principal value of a 19000° angle is \( 280^\circ \).
\(\bullet \) Principal value of the angle -19000°:
\[
\begin{array}{r|l}
-19000 & 360 \\
-19080 & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & -53\\
80 &
\end{array}
\]
The principal value of a -19000° angle is \( 80^\circ \).
Examples:
Let us find the principal values of the angles \( \frac{45\pi}{4} \) radians and \( -\frac{45\pi}{4} \) radians.
\[
\frac{45\pi}{4} = \frac{40\pi + 5\pi}{4} = 10\pi + \frac{5\pi}{4}
\]
By discarding \( 10\pi \), the principal value of the angle \( \frac{45\pi}{4} \) radians is found to be:
\[
\frac{5\pi}{4} \text{ radians.}
\]
Note that if we write:
\[
\frac{45\pi}{4} = \frac{44\pi + \pi}{4} = 11\pi + \frac{\pi}{4}
\]
and discard \( 11\pi \), we do not get the correct principal value because \( 11\pi \) is not a full integer multiple of \( 2\pi \).
For the negative angle \( -\frac{45\pi}{4} \):
\[
\begin{array}{r|l}
-45 \pi & 4 \\
-48 \pi & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & -12 \pi \\
3 \pi &
\end{array}
\]
By discarding \( -12\pi \) (which corresponds to an integer multiple divided by 4), the principal value of the angle \( -\frac{45\pi}{4} \) radians is found to be \( \frac{3\pi}{4} \) radians.
Example:
Let us find the principal value of an angle measuring 7777 grads.
\[
\begin{array}{r|l}
7777 & 400 \\
7600 & \rule{10mm}{0.30mm} \\
– \rule{15mm}{0.30mm} & 19\\
177 &
\end{array}
\]
The principal value of a 7777 grad angle is 177 grads.
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