1. WARM UP tan x = is caled a ___________ property. The value of 30 is what percentage of 1000? cos 7 cos 3 + sin 7 sin 3 = ________ What is the amplitude of the sinusoid y = 8 cos θ + 15 sin θ? If A cos (θ – D) = 8 cos θ + 15 sin θ, then D could equal Quotient 3% cos(7 – 3) = cos 4 17

2. OTHER COMPOSITE ARGUMENT PROPERTIES

3. OBJECTIVES For trigonometric functions f, derive and learn properties for: f(-x) in terms of f(x) f(90 – θ) in terms of functions of θ, or f(π/2 – x) in terms of functions of x f(A +B) and f(A – B) in terms of functions of A and functions of B

4. KEY WORDS & CONCEPTS Odd-even properties Parity complementary Cofunction properties Composite argument properties Triple argument properties

5. ODD EVEN PROPERTIES If you take the function of opposite angles or arcs, interesting patterns emerge. sin (-20°) = -0.3420…and sin 20° = 0.3420… cos (-20°) = 0.9396… andcos 20° = 0.9396… tan (-20°) = -0.3639…andtan 20° = 0.3639…

6. DEFINING MEASUREMENT OF ROTATION These numerical examples illustrate the fact that sine and tangent are opposite functions and cosine is an even function. The graphs show why these properties apply for any value of θ. The reciprocals of the function have the same parity (oddness or evenness) as the original functions.

7. PROPERTIES ODD AND EVEN FUNCTIONS Cosine and its reciprocal are even functions. That is, cos (-x) = cos xandsec (-x) = sec x Sine and tangent, and their reciprocals, are odd functions. That is, sin (-x) = -sin xandcsc (-x) = -csc x tan (-x) = -tan xandcot (-x) = -cot x

8. COFUNCTIONS PROPERTIES Properties of (90° - θ) or (π/2 – x) The angles 20° and 70° are complementary angles because they sum to 90°. (The word comes from “complete,” because the two angles complete a right angle.) The angle 20° is the complement of 70°, and the angle 70° is the complement of 20°. An interesting pattern shows up if you take the function and the co-function of complementary angles. cos 70° = 0.3420… and sin 20° = 0.3420… cot 70° = 0.3639… and tan 20° = 0.3639… csc 70° = 1.0641… and sec 20° = 1.0641…

9. PROOF You can verify these patterns by using the right triangle definitions of the trigonometric functions. The right triangle with acute angles measure 70° and 20°. The opposite leg for 70° is the adjacent leg for 20°. Thus, and

10. COFUNCTIONS The prefix co-in the names cosine, cotangent and cosecant comes from the word complement. In general, the cosine of an angle is the sine of the complement of that angle. The same property is true for cotangent and cosecant, as you can verify in the previous triangle. The co-function properties are true regardless of the measure of the angle or arc. For instance, if θ is 234°, the the complement of θ is 90° - 234°, or = -144°. cos 234° = −0.5877… and sin (90° − 234°) = sin (−144°) = −0.5877… cos 234° = sin (90° − 234°)

11. COFUNCTION PROPERTIES The cofunction properties for trigonometric functions are summarized verbally as Note that it doesn’t matter which of the two angles you consider to be “the angle” and which you consider to be the “complement. It is just as true, for example, that sin 20° = cos (90° - 20°) The cosine of an angle equals the sine of the complement of that angle. The cosine of an angle equals the sine of the complement of that angle. The cotangent of an angle equals the tangent of the complement of that angle The cotangent of an angle equals the tangent of the complement of that angle The cosecant of an angle equals the secant of the complement of that angle. The cosecant of an angle equals the secant of the complement of that angle.

12. PROPERTIES Cofunction Properties for Trigonometric Functions When working with degrees: cos θ = sin (90° − θ) and sin θ = cos (90° − θ) cot θ = tan (90° − θ) and tan θ = cot (90° − θ) csc θ = sec (90° − θ) and sec θ = csc (90° − θ) When working with radians: cos x = sin ( − x) and sin x = cos ( − x) cot x = tan ( − x) and tan x = cot ( − x) csc x = sec ( − x) and sec x = csc ( − x)

13. COMPOSITE ARGUMENT PROPERTY for cos (A + B) You can write the cosines of a sum of two angles in terms of functions of those two angles. You can transform the cosine of a sum to a cosine of a difference with some insightful algebra and the odd-even properties. cos (A + B) Change the sum into a difference. = cos [A – (-B)] Use the composite argument property for cos (first – second) = cos A cos (-B) + sin A sin (- B) Cosine is an even function. Sine is an odd function = cos A cos B + sin A (-sin B) = cos A cos B – sin A sin B The only difference between this property and the one for cos (A – B) is the sign between the terms on the right side of the equation. = cos (A + B) = cos A cos B – sin A sin B

14. COMPOSITE ARGUMENT PROPERTY for sin (A – B) and sin (A + B) You can derive composite argument properties for sin (A – B) with the help of the cofunction property. sin (A – B) = cos [90 – [A – B)] Transform into a cosine using the cofunction property = cos [(90° – A) + B] Distribute the minus sign, then associate (90° – A) = cos (90 – A )cos B – sin (90° – A) sin B Use the composite argument property for cos (first + second) = sin A cos B – cos A sin B You can derive it by writing sin (A + B) as sin [A – (-B)] and using the same reasoning as for cos (A + B). sin (A + B) = sin A cos B + cos A sin B The composite argument property for sin (A + B) is

15. COMPOSITE ARGUMENT PROPERTY for tan (A – B) and tan (A + B) You can write the tangent of a composite argument in terms of tangents of the two angles. This requires factoring out a “common” factor that isn’t actually there! Us the quotient property for tangent to bring in” sines and cosines. Use the composite argument properties for sin (A – B) and cos (a – B). Factor out (cos A cos B) in the numerator and denominator to put cosines in the minor denominator. Cancel all common factors. Use the quotient property to get only tangents.

16. SOLUTIONS You can derive it by writing sin (A + B) as sin [A – (-B)] and using the same reasoning as for cos (A + B). The box on the next page summarizes the composite argument properties for cosine, sine and tangent. As with the composite argument properties for cos (A – B) and cos (A + B), notice the signs between the terms change when you compare sin (A – B) and sin (A + B) or tan (A – B) andtan (A + B).

17. PROPERTIES Composite Argument Properties for Cosine, Sine, and Tangent cos (A – B) = cos A cos B + sin A sin B cos (A + B) = cos A cos B – sin A sin B sin (A – B) = sin A cos B – cos A sin B sin (A + B) = sin A cos B + cos A sin B

18. ALGEBRAIC SOLUTION OF EQUATIONS You can use the composite argument properties to solve certain trigonometric equations algebraically. Example 1: Solve the equation for x [0, 2π]. Verify the solutions graphically. Solution: Write the given equation. Use the composite argument property for sin (A – B)

19. SOLUTION CONTINUED = 0.5235… + 2πn In this case, the answers turn out to be simple multiples of π. See if you can figure out why x = Use the definition of arcsine to write the general solution. Use n = 0 and n = 1 to get the solutions in the domain. or (π – 0.5235…) + 2πn x = 0.2617… + πn or 1.3089… + πn x = 0.261…, 1.3089…, 3.4033…, 4.4505… 2x = arcsine 1/2

20. SOLUTION CONTINUED The graph show y = sin 5x cos 3x – cos 5x sin 3 and the line y = 0.5. Note that the grpha of y = sin 5x cos 3x – cos 5x sine 3x is equivalent to the sinusoid y – isn 2x. By using the intersect feature, you can see that th four solutions are correct and that they are the only solutions in the domain x [0, 2π]

21. CH. 5. 3 HOMEWORK Textbook pg. 211 #2-34 every other even