Principal values questions with answers

Questions

1. Question 1Principal values

   Without using a calculator, find the principal value, in degrees, of $\sin^{-1}\left(\dfrac{1}{2}\right)$.

   Show the worked solution

   1. Step 1: Find the basic angle (use the positive value)$\text{Basic angle} = 30^\circ$
   2. Step 2: Which range does the principal value of sin⁻¹ lie in?The principal value of $\sin^{-1}$ lies in $-90^\circ$ to $90^\circ$.
   3. Step 3: Use the sign of the value to place the angle in that range$\sin^{-1}\left(\dfrac{1}{2}\right) = 30^\circ$
2. Question 2Principal values

   Without using a calculator, find the principal value, in degrees, of $\cos^{-1}\left(-\dfrac{1}{2}\right)$.

   Show the worked solution

   1. Step 1: Find the basic angle (use the positive value)$\text{Basic angle} = 60^\circ$
   2. Step 2: Which range does the principal value of cos⁻¹ lie in?The principal value of $\cos^{-1}$ lies in $0^\circ$ to $180^\circ$.
   3. Step 3: Use the sign of the value to place the angle in that range$\cos^{-1}\left(-\dfrac{1}{2}\right) = 120^\circ$
3. Question 3Principal values

   Without using a calculator, find the principal value, in degrees, of $\tan^{-1}\left(\dfrac{\sqrt{3}}{3}\right)$.

   Show the worked solution

   1. Step 1: Find the basic angle (use the positive value)$\text{Basic angle} = 30^\circ$
   2. Step 2: Which range does the principal value of tan⁻¹ lie in?The principal value of $\tan^{-1}$ lies in $-90^\circ$ to $90^\circ$.
   3. Step 3: Use the sign of the value to place the angle in that range$\tan^{-1}\left(\dfrac{\sqrt{3}}{3}\right) = 30^\circ$