How To Solve Triangles

# How To Solve Triangles - Functions of Special Angles (Trig...

This preview shows pages 1–4. Sign up to view the full content.

Functions of Special Angles (Trig without Tears Part 3) OakRoadSystems Articles Mathematics Trig without Tears 3/Special Angles Shareware Utils | About Us | Site Map | Search Like this article? Please donate a few bucks . Trig without Tears Part 3: Functions of Special Angles revised 13 Jun 2006 Copyright © 1997–2007 Stan Brown, Oak Road Systems Summary: You need to know the function values of certain special angles, namely 30° ( π /6), 45° ( π /4), and 60° ( π /3) . You also need to be able to go backward and know what angle has a sine of ½ or a tangent of -√ 3. While it’s easy to work them out as you go (using easy right triangles), you really need to memorize them because you’ll use them so often that deriving them or looking them up every time would really slow you down. Functions of 45° Look at this 45-45-90° triangle, which means sides a and b are equal. By the Pythagorean theorem, a² + b² = c² But a = b and c = 1; therefore http://oakroadsystems.com/twt/special.htm (1 of 4)10/11/2007 5:05:44 PM

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Functions of Special Angles (Trig without Tears Part 3) 2a² = 1 a² = 1/2 a = 1/ 2 = ( 2)/2 Since a = sin 45°, sin 45° = ( 2)/2 Also, b = cos 45° and b = a; therefore cos 45° = ( 2)/2 Use the definition of tan A, equation 3 or equation 4 : tan 45° = a/b = 1 (14) sin 45° = cos 45° = ( 2)/2 tan 45° = 1 Functions of 30° and 60° Now look at this diagram. I’ve drawn two 30-60- 90° triangles back to back, so that the two 30° angles are next to each other. Since 2×30° = 60°, the big triangle is a 60-60-60° equilateral triangle. Each of the small triangles has hypotenuse 1, so the length 2b is also 1, which means that b = ½2s But b also equals cos 60°, and therefore cos 60° = ½ You can find a, which is sin 60°, by using the Pythagorean theorem: (½)² + a² = c² = 1 1/4 + a² = 1 a² = 3/4 a = ( 3)/2 http://oakroadsystems.com/twt/special.htm (2 of 4)10/11/2007 5:05:44 PM
Since a = sin 60°, sin 60° = ( 3)/2. Since you know the sine and cosine of 60°, you can easily use the cofunction identities ( equation 2 ) to get the cosine and sine of 30°: cos 30° = sin (90° - 30°) = sin 60° = ( 3)/2 sin 30° = cos (90° - 30°) = cos 60° = 1/2 As before, use the definition of the tangent to find the tangents of 30° and 60° from the sines and cosines: tan 30° = sin 30° / cos 30° tan 30° = (1/2) / (( 3)/2) tan 30° = 1 / 3 = ( 3)/3 and tan 60° = sin 60° / cos 60° tan 60° = (( 3)/2) / (1/2) tan 60° = 3 The values of the trig functions of 30° and 60° can be summarized like this: (15) sin 30° = ½, sin 60° = ( 3)/2 cos 30° = ( 3)/2, cos 60° = ½ tan 30° = ( 3)/3, tan 60° = 3 Mnemonic for All Special Angles Incidentally, the sines and cosines of 0, 30°, 45°, 60° and 90° display a pleasing pattern: (16) for angle A = 0, 30° ( π /6), 45° ( π /4), 60° ( π /3), 90° (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/28/2008 for the course MA 113 taught by Professor Dr.lewis during the Spring '08 term at Kentucky.

### Page1 / 16

How To Solve Triangles - Functions of Special Angles (Trig...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online