By David Eugene Smith
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Extra resources for A Source Book in Mathematics: v. 2
Between 180◦ and 360◦ ) are all negative, going to a minimum of −1 for sin 270◦ , and returning to 0 for sine 360◦ . The brain remembers pictures better than equations or words; so commit the graph on the opposite page to memory; and this would prove to be extremely valuable in solving trig problems. This graph is well-known in Math as “the sinusoidal curve” or the “sine curve”, for short. A quick sketch of the sinusoidal curve (done in 10 seconds) will provide a good guide for ensuring that the correct values of sine in the different quadrants are obtained.
While these functions were extremely useful before calculators and computers were available, they have fallen into disuse in modern times except for purposes of “examinations”, to test the student’s versatility. ∗ Have Fun! ∗ There are no simple sum and difference formulas for tan! September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) This page intentionally left blank 07˙level-3 September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) Practical Trig Numerical Values of Special Angles 07˙level-3 September 17, 2007 22:20 Book:- Trig or Treat 34 (9in x 6in) 07˙level-3 Trig or Treat Numerical Values of Special Angles 30° 30° 2 2 3 60° 60° 1 sin 30° = sin 60° = 1 1 2 cos 30° = 3 2 cos 60° = 3 2 tan 30° = 1 2 tan 60° = 1 3 2 3 1 45° 90° 1 sin 45° = 1 2 cos 45° = 1 2 tan 45° = 1 1 September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Numerical Values of Special Angles 35 Numerical Values of Special Angles Five angles are of special interest in trigonometry — 0 ◦ , 30◦ , 45◦ , 60◦ , 90◦ .
With these “12 + 1” key functions, Level-Two-Games (Less-Easy Proofs) should prove to be easy also. Within Level-Two-Games, the proofs begin with the simpler ones and progress upwards in difficulty. Have Fun! This page intentionally left blank September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) Trig — Level Three Angles in a Triangle 07˙level-3 September 17, 2007 22:20 Book:- Trig or Treat 26 (9in x 6in) 07˙level-3 Trig or Treat Special Trig Identities for All Three Angles in a Triangle A C B A B C + + = 90° 2 2 2 A + B + C = 180° C = 180° − (A + B) sin C = sin (180° − (A + B)) = sin (A + B) cos C = cos (180° − (A + B)) = −cos (A + B) ( A +2 B )) A+B = cos ( ) 2 C A+B cos = cos (90° − ( )) 2 2 sin ( C = sin 90° − 2 = sin sin (A + B + C) = sin 180° = 0 cos (A + B + C) = cos 180° = −1 See Graphs on p.