CRASH COURSE IN PRECALCULUS

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1 CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai

2 Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter 5-7.

3 LECTURE 7. TRIGONOMETRY: PART II This lecture is the second part of reviewing high school trigonometry: addition and subtraction, double and half angle, product-to-sum formulas and sum-to-product formulas for trigonometric functions and some of their applications; and area formulas and the laws of sines and cosines for general triangles.

4 Addition and Subtraction Formulas for Trig. Function sin(α + β) = sin α cos β + cos α sin β Formulas for Sine: sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β Formulas for Cosine: cos(α β) = cos α cos β + sin α sin β tan α + tan β tan(α + β) = Formulas for Tangent: 1 tan α tan β tan α tan β tan(α β) = 1 + tan α tan β

5 Some Proofs of the Formulas Proof of Subtraction Formula for Sine: By the addition formula for sine and even-odd identities, sin(α β) = sin [α + ( β)] = sin α cos( β) + cos α sin( β) = sin α cos β cos α sin β. Proof of Subtraction Formula for Cosine: By the addition formula for cosine and even-odd identities, cos(α β) = cos [α + ( β)] = cos α cos( β) sin α sin( β) = cos α cos β + sin α sin β. Proof Addition Formula for Tangent: By Addition Formula for Sine and Cosine and Reciprocal Identities, sin(α + β) tan(α + β) = cos(α + β) sin α cos β+cos α sin β cos α cos β cos α cos β sin α sin β cos α cos β = = sin α cos β + cos α sin β cos α cos β sin α sin β = tan α + tan β 1 tan α tan β

6 Examples 1. sin 7π 1 =? sin 7π 1 = sin (3π + 4π 1 ) = sin ( π 4 + π 3 ) = 1 + sin π 4 cos π 3 + cos π 4 sin π 3 =. cos π 9 cos π 9 sin π 9 sin π 9 =? 3 = (1 + 3). 4 cos π 9 cos π 9 sin π 9 sin π 9 = cos (π 9 + π 9 ) = cos π 3 = Prove the identity 1 + tan θ 1 tan θ = tan (π 4 + θ). 1 + tan θ 1 tan θ = tan π 4 + tan θ 1 tan π 4 tan θ = tan (π 4 + θ).

7 Examples 4. Express sin(cos 1 x + tan 1 y) as an algebraic expression in x and y, where x [ 1, 1] and y R. Let α = cos 1 x and β = tan 1 y. Then cos α = x, tan β = y and sin(cos 1 x + tan 1 y) = sin(α + β) = sin α cos β + cos α sin β. cos α = x sin α = 1 x. y tan β = y sin β = and cos β = 1. Hence 1+y 1+y sin(cos 1 x + tan 1 y) = 1 x 1 + x y = 1 + y 1 + y 1 x + xy 1 + y = ( 1 x + xy) 1 + y 1 + y.

8 Examples 5. Use the addition and subtraction for sine to simplify 1 3 sin θ + cos θ in terms of a single trigonometric function. 1 3 sin θ + cos θ = cos π 3 sin θ + sin π 3 cos θ = sin (θ + π 3 ). On the other hand, 1 3 sin θ + cos θ = sin π 6 sin θ + cos π 6 cos θ = sin (θ π 6 ). More generally, we can write, for any A, B R with A + B 0, A cos θ + B sin θ = A + B A ( A + B cos θ + B sin θ) A + B = A + B (cos ϕ 1 cos θ + sin ϕ 1 sin ϕ 1 ) = A + B cos(θ ϕ 1 ) = A + B (sin ϕ cos θ + cos ϕ sin ϕ ) = A + B sin(θ + ϕ )

9 Sum of Sines and Cosines For any A, B R with A + B 0 A cos θ + B sin θ = A + B cos(θ ϕ 1 ) = A + B sin(θ + ϕ ) cos ϕ 1 = where sin ϕ 1 = A A +B B A +B and sin ϕ = cos ϕ = A A +B B A +B We note that by knowing the values any two distinct trigonometric functions of the six trigonometric functions, as long as they are not from the three reciprocal identities, the values of remaining four trigonometric functions are also determined.

10 An Example Prove the identity cos θ sin θ cos θ + sin θ = tan (π 4 θ). cos θ sin θ ( cos θ + sin θ = cos θ sin θ) = ( cos θ + sin θ) cos θ sin θ cos θ + sin θ = sin π 4 cos θ cos π 4 sin θ cos π 4 cos θ + sin π 4 sin θ = sin ( π 4 θ) cos ( π 4 θ) = tan (π 4 θ)

11 Double Angle Formulas for Trigonometric Functions With α = β = θ in the addition formulas for sine, cosine and tangent functions, we have Formula for Sine Function: Formula for Cosine Function: sin θ = sin θ cos θ cos θ sin θ cos θ = 1 sin θ cos θ 1 Formula for Tangent Function: tan θ = tan θ 1 tan θ.

12 Half Angle Formula for Sine and Cosine Functions 1 sin θ Using cos θ = cos θ 1, replacing θ, θ by θ, θ correspondingly and rearrange terms, we obtain and so sin θ = 1 cos θ, cos θ = 1 + cos θ sin θ 1 cos θ = ±, cos θ 1 + cos θ = ± The choice of + or sign depends on the quadrant in which θ lies.

13 Half Angle Formula for Tangent Function Using tan θ = tan θ 1 tan θ tan θ = tan θ 1 tan θ t 1 t, where t = tan θ, and = we choose b = t and a = 1 t, so c = (t) + (1 t ) = 4t + 1 t + t 4 = 1 + t + t 4 = (1 + t ) = 1 + t. Therefore,

14 Half Angle Formula for Tangent Function sin θ = tan θ 1 + tan θ cos θ = 1 tan θ 1 + tan θ tan θ = tan θ 1 tan θ csc θ = 1 + tan θ tan θ sec θ = 1 + tan θ 1 tan θ cot θ = 1 tan θ tan θ

15 Examples 1. sin π 1 =? Note that sin π π 1 = sin 6 and π 1 we see that. tan 7π 8 =? sin π 1 = 1 cos π 6 = lies in the first quadrant, 1 3 = 3 tan 7π 8 = tan (π + 3π 8 ) = tan [π ( 3π 8 )] = cot ( 3π 8 ) = cot 3π 8 = cos 3π 8 sin 3π 8 Hence tan 7π 8 = and 3π 3π 8 = 4 1+cos 3π 4 1 cos 3π 4 is in the the first quadrant. = 1 = =

16 Examples 3. Write sin( cos 1 x) as an algebraic expression in x only, where x [ 1, 1]. Let θ = cos 1 x. Then cos θ = x and sin( cos 1 x) = sin θ = sin θ cos θ. Using we have sin θ = 1 x. Hence sin( cos 1 x) = x 1 x.

17 Product-to-Sum Formulas sin(α + β) = sin α cos β + cos α sin β Idea: Recalling sin(α β) = sin α cos β cos α sin β and adding the left- and right-sides of these formulas, gives sin(α + β) + sin(α β) = sin α cos β So sin(α + β) + sin(α β) sin α cos β = Similarly, by subtracting them on both sides, gives cos α sin β = sin(α + β) sin(α β) Apply similar techniques to the addition and subtraction formulas of cosine function to see

18 Product-to-Sum Formulas sin(α + β) + sin(α β) sin α cos β = sin(α + β) sin(α β) cos α sin β = cos(α + β) + cos(α β) cos α cos β = cos(α β) sin(α + β) sin α sin β =

19 Examples 1. Express sin 3x cos 5x as a sum of trigonometric functions. = sin 3x cos 5x = = sin 8x + sin( x) sin(3x + 5x) + sin(3x 5x) = sin 8x sin x sin 3x cos 5x = cos 5x sin 3x sin(5x + 3x) sin(5x 3x) = or sin 8x sin x. Express sin 3x sin 5x as a sum of trigonometric functions. sin 3x sin 5x = = cos( x) cos 8x cos(3x 5x) cos(3x 5x) = cos x cos 8x

20 Sum-to-Product Formulas By applying the Product-to-Sum formulas, with α = x+y x = α + β β = x y, and rearranging terms, we obtain y = α β sin x + sin y = sin x + y cos x y sin x sin y = cos x + y sin x y cos x + cos y = cos x + y cos x y cos x cos y = sin x + y sin x y

21 Examples 1. Write sin 7x + sin 3x as a multiple of trigonometric functions. 7x + 3x 7x 3x sin 7x + sin 3x = sin cos = sin 5x cos x. sin 3x sin x. Simply the fractional expression cos 3x + cos x. 3x+x sin 3x sin x cos sin 3x x cos x sin x = cos 3x + cos x cos 3x+x cos 3x x = cos x cos x = sin x = tan x. cos x

22 Area Formulas for General Triangles Given a triangle with side lengths a and b, and included angle θ, then the area A = 1 ab sin θ

23 Area Formulas for General Triangles Heron s Formula: A = s(s a)(s b)(s c) where s = a + b + c is the semiperimeter of the triangle; that is half of the perimeter.

24 Laws of Sines and Cosines for General Triangles Law of Sines: In ABC we have sin α a Law of Cosines: In ABC = sin β b = sin γ c a = b + c bc cos α b = a + c ac cos β c = a + b ab cos γ

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