Question

Determine which of the following pairs of angles are coterminal.
A.\[210^\circ , - 150^\circ \]
B.\[330^\circ , - 60^\circ \]
C.\[405^\circ , - 675^\circ \]
D.\[1230^\circ , - 930^\circ \]

Answer 1

Hint: Here, we have to find the pair of angles which are co-terminal to each other. Co-terminal angles are angles in standard position i.e., the angles with the initial side on the positive \[x\] –axis that have a common terminal side.

Formula used: We will use the formula Co-terminal of \[\theta = \theta + 360^\circ \times k\], if \[\theta \] is given in degrees.

Complete step-by-step answer:
We have to find which pair of angles are co-terminal to each other.
First, we have to find whether \[210^\circ , - 150^\circ \] are co-terminal to each other or not
By rewriting the formula Co-terminal of \[\theta = \theta + 360^\circ \times k\] , we get
\[360^\circ \times k = \] Co-terminal of \[\theta - \theta \] …………………………………………….\[\left( 1 \right)\]
Substituting the given angles in the equation \[\left( 1 \right)\], we have
\[ \Rightarrow 360^\circ \times k = 210^\circ - ( - 150^\circ )\]
We know that the multiplication of two negative integer is positive. So, we get
 \[ \Rightarrow 360^\circ \times k = 210^\circ + 150^\circ \]
Adding the terms, we get
\[ \Rightarrow 360^\circ \times k = 360^\circ \]
\[ \Rightarrow 360^\circ \times k = 360^\circ \times 1\]
Since the given pair can be expressed as \[360^\circ \times k\]
So, the given pair of angles are co-terminal to each other.
Now, we have to find whether\[330^\circ , - 60^\circ \] are co-terminal to each other or not.
Substituting the given angles in the equation \[\left( 1 \right)\], we have
\[ \Rightarrow 360^\circ \times k = 330^\circ - ( - 60^\circ )\]
 \[ \Rightarrow 360^\circ \times k = 330^\circ + 60^\circ \]
Adding the terms, we have
\[ \Rightarrow 360^\circ \times k = 390^\circ \]
Since the given pair cannot be expressed as \[360^\circ \times k\]
So, the given pair of angles are not co-terminal to each other.
Now, we have to find \[405^\circ , - 675^\circ \] are co-terminal to each other.
Substituting the given angles in the equation \[\left( 1 \right)\], we have
\[ \Rightarrow 360^\circ \times k = 405^\circ - ( - 675^\circ )\]
 \[ \Rightarrow 360^\circ \times k = 405^\circ + 675^\circ \]
Adding the terms, we have
\[ \Rightarrow 360^\circ \times k = 1080^\circ \]
\[ \Rightarrow 360^\circ \times k = 360^\circ \times 3\]
Since the given pair can be expressed as \[360^\circ \times k\]
So, the given pair of angles are co-terminal to each other.
Now, we have to find if \[1230^\circ , - 930^\circ \] are co-terminal to each other or not.
Substituting the given angles in the equation \[\left( 1 \right)\], we have
\[ \Rightarrow 360^\circ \times k = 1230^\circ - ( - 930^\circ )\]
The multiplication of two negative integers is positive.
 \[ \Rightarrow 360^\circ \times k = 1230^\circ + 930^\circ \]
Adding the terms, we have
\[ \Rightarrow 360^\circ \times k = 2160^\circ \]
\[ \Rightarrow 360^\circ \times k = 360^\circ \times 6\]
Since the given pair can be expressed as \[360^\circ \times k\]
So, the given pair of angles are co-terminal to each other.
Therefore, \[(210^\circ , - 150^\circ ),(405^\circ , - 675^\circ )\] and \[(1230^\circ , - 930^\circ )\] are co-terminal to each other.

Note: We can check whether the given angles are co-terminal by adding or subtracting the given angles. If the answer obtained is a multiple of \[360^\circ \] then the pair of angles are co-terminal. If it is not a multiple of \[360^\circ \], then the pair of angles are not co-terminal. Increasing or decreasing the degree measure of an angle in a standard position, by an integer multiple of \[360^\circ \], results in a co-terminal angle.