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8.5 Solving More Difficult Trig Equations
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8.5 Solving Trig Equations
Objectives: Solve basic trig equations Use factoring and trig identities to solve complicated trig equations Vocabulary: sine, cosine, tangent, cosecant, secant, cotangent, cofunction
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8.5 Solving Trig Equations
WARNING!!!!! Do not divide by sin, cos, tan, sec, csc or cot! You may lose answers. Always check your solutions
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8.5 Solving Trig Equations Cont
Objective to solve simple trigonometric equations and to apply them.
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SOLVE: 3 1 =2π πππ 3 2 =π πππ π ππ π 3 = 3 2 π= π 3 , 2π 3
1. Isolate the trig functions using algebra and identities 3 1 =2π πππ 3 2 =π πππ 2. Determine the quadrants Sin + in first and second π ππ π 3 = 3. Find the angle measure 4. Use reference angles to determine solutions π= π 3 , 2π 3
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Solve cos ΞΈ + 1 = sin ΞΈ in [0, 2ο°]
πππππ π ππ 2 + πππ 2 =1 π ππ= 1 β πππ 2 Like solving algebraic equation, once we squared the original trigonometric equation, it may generate some extraneous solution. We need to check the solutions. The only solutions are ο°/2 and ο°.
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If the values that these trig functions equal are NOT exact values on the unit circle you will need to use your calculator. from calculator: this value is somewhere in Quadrant I What other quadrant would have the same cosine value (same x value on the unit circle)? Quadrant IV This angle is 2ο° minus angle from calculator.
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Solve tan2x β 3tanx β 4 = 0 in 0 β€ x < 360.
tan x β 4 = 0 or tan x + 1 = 0 tan x = 4 or tan x = β1 x = arctan(4) or x = arctan(-1) x = 76Β° or x = 135Β° Quad check? x = 256Β° or x = 315Β°
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Solve 8 sin ο± = 3 cos2 ο± with ο± in the interval [0, 2Ο].
[Rewrite the equation in terms of only one trigonometric function. 8 sinο± = 3(1ο sin2ο± ) Use the Pythagorean Identity. 3 sin2ο± + 8 sinο± ο 3 = 0. A βquadraticβ equation with sin x as the variable (3 sinο± ο 1)(sinο± + 3) = 0 Factor. Therefore, 3 sinο± ο 1 = 0 or sinο± + 3 = 0 Solutions: sinο± = or sinο± = -3 1 3 ο± = arcsin( ) ο» and ο± = Ο ο arcsin( ) ο» 1 3
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Solve. 3sinx + 4 = -1/sinx in 0 β€ x β€ 2ο° .
Let x = sin x, then 3x + 4 = -1/x x(3x + 4) = x(-1/x) 3x2 + 4x = -1 (3x + 1)(x + 1) = 0 3x + 1= 0 or x + 1 = 0 x = -1/3 or x = β1 sin x = -1/3 or sin x = β1 x = ο° + arcsin(1/3) or x = 3ο°/2 x = 2ο° β arcsin(1/3)
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Solve 5cos2ο± + cos ο± β 3 = 0 for 0 β€ ο± β€ Ο.
The equation is quadratic. Let u = cosο± and solve 5u2 + u ο 3 = 0. u = -1 Β± = or 10 Therefore, cosο± = or β Use the calculator to find values of ο± in 0 β€ ο± β€ Ο. This is the range of the inverse cosine function. The solutions are: ο± = cos ο1( ) = and ο± = cos ο1(ο ) =
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Since we squared the original equation we have to check our answer.
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Since we squared the original equation we have to check our answer.
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Inclination and Slope The inclination of a line is the angle ο‘, where 0 ο£ ο‘ < ο° (0o ο£ ο‘ < 180o), that is measured from the positive x-axis to the line. The line at the left above has positive slope and the right one has negative slope. How does the slope connect to the trigonometric functions?
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Since line l2 is parallel to line l1, thus,
Suppose that the inclination of line l2 is ο‘, the line l1 is passes through the origin and is parallel to l2 . So the inclination of line l1 is ο‘. Let P be the point where l1 intersects the unit circle. Then P (cosο‘, sinο‘). The slope of line l1 can be calculated between point P and the origin: y P (cosο‘, sinο‘) r = 1 ο‘ ο‘ x l2 l1 Since line l2 is parallel to line l1, thus,
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Example Find the inclination of line 2x + 5y = 15
[Solution] Rewrite the equation as y = β2/5x + 3 Then the slope is m = β2/5 = tanο‘ Therefore ο‘ = arctan (β2/5) ο» β21.8o The reference angle is ο‘ = arctan (2/5) ο» 21.8o. The inclination is ο° β arctan(2/5) = 180o β 21.8o ο» 158.2o
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