3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

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1 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle so that every point (x,y on the circle is the cosine and sine of angles in standard position (with the initial side on the positive x-axis and the terminal side with a point somewhere on the unit circle. P (cos α, sin α (0, α-β α β P (cos β, (,0 We will use the distance formula to start proving right side of the cos(α-β equation. The distance from P to P (using the distance formula is d(p,p ( x x ( y y (cosα (sinα

2 α-β Now to get the left side of the equation, let s rotate that pink triangle so that it has one side on the positive x-axis. Page (0, P 3 (cos α-β, sin α-β A (,0 Since this is the same triangle as the previous one (just rotated, then the distance from P to P on the previous triangle is the same as the distance from P 3 to point A on this triangle. Using the distance formula where the first point is (,0 and the second point is (cos α-β, sin α-β we get: ( cos( α β ( sin( α 0 d(a,p3 β Setting those two distance equations equal to each other {d(a,p3 d(p,p } we get: ( cos( α β ( sin( α β 0 (cosα (sinα

3 Page 3 Now we can square both sides: ( cos( α β ( sin( α β 0 (cosα (sinα Now multiply out the squared terms cos ( α β cos( α β sin ( α β cos α cosα cos β cos β sin α sinα sin β Notice with these cos and sin terms we can apply the Pythagorean identity: cos θ sin θ On the left side, let θ α-β and we get: cos( α cos( α β β On the right side, let θ α, and also θ β and we get: cosα cos β sinα cosα cos β sinα Setting the right side and left side equal to each other we get; cos( α β cosα cos β sinα Now subtracting from both sides and then dividing both sides by - we get; cos(α-β cos α cos β sin α We can now prove cos(αβ cos α cos β -sin α Use the cos(α-β formula but instead of α-β, use α-(-β cos(α-(-β cos(αβ cos α cos (-β sin α sin(-β Now our even-odd properties [cos (-θ cos θ, sin(- θ -sin θ] will become useful: cos(αβ cos α cos (β - sin α sinβ

4 Now if we are asked to find the exact value of a trig function of an Angle, we can perhaps combine two known angles to get an exact answer Remember that your calculator only gives approximate values. Which angles to we know the exact values of their trig functions? (Look at your handy Unit Circle Diagram θ 0 (0 or π rad, 30 (π/6 rad, 45 (π/4 rad, 60 (π/3 rad, 90 (π/ rad and so on.. Example Find the exact value of cos is not on our Unit Circle Diagram, but we can use the fact that , and we know the trig functions of 30 and 45. cos(30 45 cos30 cos45 -sin30 sin 45 Page 4 Now you do # 5 on p. 3

5 We can use the Complementary Angle Theorem to find sin(α-β and sin(αβ Page 5 Remember: cos(π/ θ sin(θ sin(π/ θ cos(θ sin(αβ cos(π/ (αβ cos((π/ α β Shift the parentheses cos (π/ α cosβ -sin (π/ α sin α cos β -cos α Use the cos(α-β formula. Now just use the sin(αβ for sin(α-β by using this: sin(α-β sin(α (-β sin α cos (-β - cos α sin (-β And using our even-odd properties again, we get: sin(α-β sin α cosβ -cos α (- sin α cosβ cos α Now let s do Examples 3, 5 Now you do #3 a, b, and c

6 Page 6 Formulas fortan(αβ and tan(α-β Proof: tan(αβ sin( α cos( α β β sinα cos β cosα cosb cosα sinα Now divide both the numerator and denominator by cos α cos β. tan ( α β ( sinα sinα cosα sinα cosα ( sinα sin cos sin cos β β β β tan β tan β As before, we can use even-odd properties to derive the other formula.. tan(α-β tan(α(-β tan( β tan( β Recall that tan is odd so tan(-θ -tan θ, so we get: tan( α tan β β Now do #3d tan β

7 SUMMARY OF SUM AND DIFFERENCE FORMULAS cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α sin(αβ sin α cos β -cos α sin(α-β sin α cosβ cos α Page 7 tan ( α β tan β tan β tan( α β tan β tan β Example 9 on p.5 sin cos sin 3 5 α β Remember these are angles

8 Page 8 HOMEWORK p. 53 #9 36 ETP #39-59 EOO #66-75 ETP

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(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0

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