Trigonometry—Applying ASTC, Reference, Coterminal Angles

1. S A T C Trigonometry—Applying ASTC, Reference, Coterminal Angles Page15 In the old days, when people didn’t have a calculator and only had a table of reference for the trigonometric function values of angles from 0° to 90°, it’s crucial to know how to express the function in terms of a positive acute angle (i.e., its reference angle). Express as a function of a positive acute angle. 1. sin 123° = 2. cos 234° = 3. tan –345° = 4. sin 456° = 5. cos 12,345° = 6. tan –6,789° = 30° How to find the sine, cosine, tangent of special angles without a calculator: A special angle is an angle that are multiples of 30, 45 and 60. Examples: 1. sin 120 = 2. cos 135 = 3. tan 225 = 4. sin 210 = 5. cos 300 = 6. tan 330 = 60° 45° 45°

2. Hypotenuse Opposite leg  Adjacent leg Trigonometry—Secant, Cosecant and Cotangent Page16 The Other Three Functions We have learned the 3 basic definitions—sine, cosine and tangent. There are 3 more definitions—secant (sec), cosecant (csc) and cotangent (cot). The basic three: The other three: It turns out most people don’t remember the definitions of secant, cosecant and cotangent, instead, they remember them as the ____________ of the basic three. Q: How can we memorize the other three as the reciprocals the basic three without mixing them up? A: Just remember they pair each other with a different initial letter: tan  sec  csc 

3. S A T C Trigonometry—Secant, Cosecant and Cotangent (cont’d) Page17 If  is in Q I, we know: sin  = (+), cos  = (+), tan  = (+). What about ... csc  = ( ), sec  = ( ), cot  = ( ) If  is in Q II, we know: sin  = (+), cos  = (–), tan  = (–). What about ... csc  = ( ), sec  = ( ), cot  = ( ) If  is in Q III, we know: sin  = (–), cos  = (–), tan  = (+). What about ... csc  = ( ), sec  = ( ), cot  = ( ) If  is in Q IV, we know: sin  = (–), cos  = (+), tan  = (–). What about ... csc  = ( ), sec  = ( ), cot  = ( ) How to use calculator to compute cosecant, secant and cotangent: Unlike sin, cos and tan, most (maybe all) calculators do not have the built-in functions for csc, sec and cot. In order to evaluate, for example, csc 41, one must use sin key in the calculator by entering: 1  sin 41 (since csc 41 = 1/sin 41). The only time one should not use a calculator is when the angle is a special angle, i.e., 30, 45, 60 and multiples of these angles. Examples: 1. csc 41 = 2. cot 65 = 3. sec 32 = 4. csc 45 = 5. cot 60 = 6. sec 30 =

4. 90°– r y  x Trigonometry—Introduction to Identities Page18 Identities When two sides of an equation are equal regardless of what the value for the variable is, it’s called an identity. For example, x + x = 2x is an identity. In trigonometry, we have many identities, one of which we have seen already: Reciprocal Identities Other identities includes (concluded from the right triangle above): Cofunction IdentitiesQuotient Identities

5. r y  x Trigonometry—sin 2 + cos 2 = 1 Page19 One Important Identity: sin 2 + cos 2 = 1 Let see if it’s true when  = 45°, 120°, 36789°. sin 2 45° + cos 2 45° = 1 sin 2 120° + cos 2 120° = 1 sin 2 36789° + cos 2 36789° = 1 ? ? ? Why is always true? Write an equation that connects , y and r: Write an equation that connects , x and r: Write an equation that connects y, x and r: sin 2  + cos 2  = 1 Pythagorean Identities: tan 2  + 1 = sec 2  1 + cot 2  = csc 2 

6. Trigonometry—Proving Identities (cont’d) Page20 So far, we’ve learned some basic identities: Cofunction Identities and Reciprocal Identities Quotient Identities Pythagorean Identities sin 2  + cos 2  = 1 tan 2  + 1 = sec 2  1 + cot 2  = csc 2  We will learn more fundamental identities, but in the mean time, let’s use these ones to prove some identities that are not so obvious: 1. 2.

7. Trigonometry—Proving Identities Page21 Basic Trigonometric Identities: Reciprocal Identities Quotient Identities Pythagorean Identities sin 2  + cos 2  = 1 tan 2  + 1 = sec 2  1 + cot 2  = csc 2  Techniques of proving identities: 1. Use the basic identities wisely. – Change everything in terms of sine and/or cosine – If you see (trig. function)2, it might have to do with Pyth. Id. – Treat an identity in different ways: sin 2  + cos 2  = 1  sin 2  = 1 – cos 2  = 1  _________________ 2. Try to keep one side the same, and only simplify the complicated side or the side you can do something with. 3. If you need to change both sides, change each side independently. 4. Basic algebraic manipulation still apply—FOILing, expanding, factoring, simplifying, rationalizing denominator, and combining two fractional expressions into one. 5. The goal is to show left hand side = right hand side. Ex. 3:

8.  10 x  30° 12 B c a C A b Trigonometry—Law of Cosines Page22 Law of Cosines: c2 = a2 + b2 – 2abcos C This side and this angle are opposite of each other, and one of them should be the unknown To find x (of the triangle shown on the right), we use the Law of Cosines by setting up the equation as follows: x2 = The Law of Cosines (LoC) not only can be used to find an unknown side, it can be also used to find an unknown angle. For example, we can use LoC to find  too: Why is true? Let’s prove it:

9. B c a h C A b 60° 11 7 8 5 6 30° 10 9 Trigonometry—2nd Formula for the Area of the Triangle Page23 Area of a Triangle: A = ½absin C (derived from A = ½bh, of course) Proof: In words, the area of a triangle is _____________________________________________ Problems: 1. 2. 3.