Lesson 3

Using Special Triangles to Find Angles

In Try This 2 you saw how cosine, sine, and tangent can be determined for a 45° angle without using technology. Watch Finding Trigonometric Ratios for 30° and 60° Angles Without Technology.

You can now create a very basic table of trigonometric values based on the ratios found using

• angles with terminal arms that lie on the x- or y-axis of a circle (0°, 90°, 180°, 270°, and 360°)
  
  
• angles found in special triangles (30°, 45°, 60°)

θ	cos θ	sin θ	tan θ
0°	1	0	0
30°
45°			1
60°
90°	0	1	undefined
180°	−1	0	0
270°	0	−1	undefined
360°	1	0	0

So far you have used these special triangles only for angles in the first quadrant (0° ≤ θ ≤ 90°). These triangles can be used to find angles in other quadrants too. The key is to use one of these special triangles to determine the coordinates of a point on the terminal arm of the angle in question. Watch Using a 30-60-90 Triangle for a 150° Angle.

Sometimes it’s not as obvious to see how to use one of the special triangles. Watch Using a 30-60-90 Triangle for a 300° Angle to see how a similar process can be used for a 300° angle.