Exact values questions with answers

Questions

1. Question 1One special angle

   Without using a calculator, find the exact value of $\sin 30^\circ$.

   Show the worked solution

   1. Step 1: Use the special triangles (half a square, half an equilateral triangle)$\sin 30^\circ = \dfrac{1}{2}$
2. Question 2One special angle

   Without using a calculator, find the exact value of $\tan 60^\circ$.

   Show the worked solution

   1. Step 1: Use the special triangles (half a square, half an equilateral triangle)$\tan 60^\circ = \sqrt{3}$
3. Question 3Combine exact values

   Without using a calculator, find the exact value of $\tan 60^\circ \cos 30^\circ$.

   Show the worked solution

   1. Step 1: Write each exact value$\tan 60^\circ = \sqrt{3}$$\cos 30^\circ = \dfrac{\sqrt{3}}{2}$
   2. Step 2: Put the values in and multiply$\tan 60^\circ \times \cos 30^\circ = \left(\sqrt{3}\right) \times \left(\dfrac{\sqrt{3}}{2}\right)$$= \dfrac{3}{2}$
4. Question 4Combine exact values

   Without using a calculator, find the exact value of $\cos 45^\circ - \sin 45^\circ$.

   Show the worked solution

   1. Step 1: Write each exact value$\cos 45^\circ = \dfrac{\sqrt{2}}{2}$$\sin 45^\circ = \dfrac{\sqrt{2}}{2}$
   2. Step 2: Put the values in and subtract$\cos 45^\circ - \sin 45^\circ = \left(\dfrac{\sqrt{2}}{2}\right) - \left(\dfrac{\sqrt{2}}{2}\right)$$= 0$
5. Question 5In radians

   Without using a calculator, find the exact value of $\sin \dfrac{\pi}{6} \cos \dfrac{\pi}{3}$.

   Show the worked solution

   1. Step 1: Write each exact value$\sin \dfrac{\pi}{6} = \dfrac{1}{2}$$\cos \dfrac{\pi}{3} = \dfrac{1}{2}$
   2. Step 2: Put the values in and multiply$\sin \dfrac{\pi}{6} \times \cos \dfrac{\pi}{3} = \left(\dfrac{1}{2}\right) \times \left(\dfrac{1}{2}\right)$$= \dfrac{1}{4}$
6. Question 6In radians

   Without using a calculator, find the exact value of $\sin \dfrac{\pi}{6}$.

   Show the worked solution

   1. Step 1: π radians = 180°: change to degrees first$\dfrac{\pi}{6} = \dfrac{180^\circ}{6}$$= 30^\circ$
   2. Step 2: Use the special triangles (half a square, half an equilateral triangle)$\sin \dfrac{\pi}{6} = \sin 30^\circ$$\phantom{\sin \dfrac{\pi}{6}} = \dfrac{1}{2}$