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Essential Question: What are the two methods that can be used to solve a trigonometric equations graphically? (Hint: You’ve already seen them)

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8-1: Graphical Solutions to Trig Eq We did similar work in section 2.1 (that’s first marking period, folks!), so hopefully some of this will seem familiar. We can solve basic trigonometric equations graphically by two ways, using the Intersection method and the x-Intercept method

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8-1: Graphical Solutions to Trig Eq The Intersection Method Example: Solve tan x = 2 Create a graph with two equations y1 = tan x y2 = 2 Tan completes its cycle from (- / 2, / 2 ), so find any intersection in this interval G RAPH -> M ORE -> M ATH -> M ORE -> I SECT One solution is present, x = 1.1071 Because the function “tan x” has a period of , we say the solution is x 1.1071 + k

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8-1: Graphical Solutions to Trig Eq The x-Intercept Method Example: Solve sin x = -0.75 Rewrite the equation to get the right side = 0 sin x + 0.75 = 0 Sin completes its cycle from [0, 2 ], so find any intersection(s) in this interval G RAPH -> M ORE -> M ATH -> R OOT Two solutions are present, x = 3.9897 and x = 5.4351 Because the function “sin x” has a period of 2 , we say the solutions are x 3.9897 + 2 k and x 5.4351 + 2 k

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8-1: Graphical Solutions to Trig Eq Equations with multiple trigonometric functions Example: 3 sin 2 x – cos x – 2 = 0 Determine the period of each function. Use the largest period of the group. 3 sin 2 x has a period of 2 cos x has a period of 2 Use the period 2 (~6.28) to find solutions Use the root method (equations is “= 0”) 4 Solutions x 1.1216 + 2 k x 2.4459 + 2 k x 3.8373 + 2 k x 5.1616 + 2 k

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8-1: Graphical Solutions to Trig Eq Equations with multiple functions & multiple periods Example: tan x – 0.5 = 3 sin x Period of tan x – 0.5 is Period of 3 sin x is 2 Use a period of 2 for finding solutions Use the intersection method to find solutions (equations on each side) 4 Solutions x 1.2829 + 2 k x 3.2667 + 2 k x 5.1324 + 2 k x 6.0260 + 2 k

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8-1: Graphical Solutions to Trig Eq We can solve equations in degree mode, but one thing must be changed G RAPH -> W IND -> X M AX must be set to 360 Reason: When talking about degrees, we can get any value from 0 to 360 Example: Find all angles with 0 < < 360 that are solutions to: 2 sin 2 - 3 sin - 3 = 0 Graph is on the next slide

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8-1: Graphical Solutions to Trig Eq 2 sin 2 - 3 sin - 3 = 0 Use the x-Intercept method Two solutions x 223.33˚ x 316.67˚

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8-1: Graphical Solutions to Trig Eq Assignment Page 528 Problems 1 – 11 & 19 – 27 (odds) The first set is all in radian mode, the second in degree mode Remember: sin 2 x must be entered in the calculator as (sin x) 2 When you are finished doing the problems in degree mode, remember to set your xMax back to 10

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