Ratios of any angle questions with answers

Questions

1. Question 1The basic angle

   Find the basic angle of $290^\circ$.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 4$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 360^\circ - 290^\circ$$\alpha = 70^\circ$
2. Question 2The basic angle

   Find the basic angle of $98^\circ$.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 2$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 180^\circ - 98^\circ$$\alpha = 82^\circ$
3. Question 3Exact values of any angle

   Without using a calculator, find the exact value of $\cos 300^\circ$.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 4$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 360^\circ - 300^\circ$$\alpha = 60^\circ$
   3. Step 3: Is cos positive or negative in that quadrant?$\cos 300^\circ = \cos 60^\circ$$= \dfrac{1}{2}$
4. Question 4Exact values of any angle

   Without using a calculator, find the exact value of $\sin 225^\circ$.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 3$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 225^\circ - 180^\circ$$\alpha = 45^\circ$
   3. Step 3: Is sin positive or negative in that quadrant?$\sin 225^\circ = -\sin 45^\circ$$= -\dfrac{\sqrt{2}}{2}$
5. Question 5As ± the ratio of the basic angle

   Express $\tan 340^\circ$ in terms of a trigonometric ratio of its basic angle.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 4$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 360^\circ - 340^\circ$$\alpha = 20^\circ$
   3. Step 3: Is tan positive or negative in that quadrant?$\tan 340^\circ = -\tan 20^\circ$
6. Question 6As ± the ratio of the basic angle

   Express $\tan 200^\circ$ in terms of a trigonometric ratio of its basic angle.

   Show the worked solution

   1. Step 1: Which quadrant is the angle in?$\text{Quadrant} = 3$
   2. Step 2: The basic angle is the acute angle to the x-axis$\mathstrut \alpha = 200^\circ - 180^\circ$$\alpha = 20^\circ$
   3. Step 3: Is tan positive or negative in that quadrant?$\tan 200^\circ = \tan 20^\circ$