Tech Math 2 Notes Section 20.1 Page 1 of 3 Section 20.1: Fundamental Trigonometric Identities Big Idea: Expressions containing trig functions can be simplified a lot of times using identities that replace multiple functions with fewer functions. Big Skill: You should be able to simplify trigonometric expressions using the identities in this section. Starting point: For Lab #1, some people got different formulas for s1: However, both these formulas give the same, correct answers, because for a 6/12 pitch, = tan-1(6/12) 26.57, and sin(26.57) 0.4473, and cos(90 - 26.57) = cos(63.43) 0.4473. Thus, we have seen that sin cos 90 . This kind of relationship is true in general as follows, and all these relationships are called “the complementary angle identities” sin 90 cos Complementary Angle Identities tan 90 cot sec 90 csc cos 90 sin cot 90 tan csc 90 sec In fact, the “co” part of the cosine, cotangent, and cosecant is short for complementary. Tech Math 2 Notes Section 20.1 Page 2 of 3 Next: how come I don’t have buttons on my calculator for cotangent, secant, and cosecant? Look at the following right triangle and compute side b: opp b b sin 20 hyp 1.000 We can also get b using the cosecant function: hyp 1.000 1 csc 20 b opp b csc 20 sin 20 Since these both give the correct formula for b, it must be true that: sin 20 as: csc 20 1 . sin 20 This is an example of one of the reciprocal identities. Since sec tan 1 , which can also be written csc 20 hyp adj adj and cos , and cot and adj opp hyp opp , we can get two more reciprocal identities: adj 1 csc sin Reciprocal Identities 1 sec cos cot 1 tan Looking at the same right triangle: opp sin 20 hyp opp adj opp tan 20 . This gives us the ratio identities: and cos 20 , so sin 20 cos 20 adj adj hyp hyp hyp Ratio Identities sin cos tan cot cos sin Tech Math 2 Notes Section 20.1 Page 3 of 3 Looking at the same right triangle a third time: The Pythagorean Theorem tells us that: a2 + b2 = c2. But: b = sin 20, and a = cos 20. Plugging that into the Pythagorean Theorem: (cos 20)2 + (sin 20)2 = 1 This equation is true for any angle, and leads to the Pythagorean Identities: Pythagorean Identities 2 2 sin cos 1 1 tan 2 sec 2 sin 2 1 cos 2 tan 2 sec 2 1 1 cot 2 csc 2 cot 2 csc 2 1 cos 2 1 sin 2 So, now that we have all these identities, what are they good for? They are good for simplifying formulas that have trigonometric functions. Example: w 0.5w sin . Show that this simplifies to: From Lab #1, we got L7 0.5 L tan cos cos w L7 0.5 tan L w . cos La dee dah… Tips for Simplifying Trigonometric Expressions or Proving Trigonometric Identities: Sometimes it helps to write everything as sines and cosines. Look for pieces that can be simplified using the Pythagorean Identities. Sometimes it helps to multiply things out. Sometimes it helps to factor things. Sometimes you’ll have to find common denominators to add fractions.