Download MATH 1114 Test #2

MATH 1114 Test #3
7.7 Solving Trigonometric Equations I
( text: 7, 13, 31, 41; others 9, 11, 21, 25, 29, 33, 37, 39, 53)
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Solving Equations Involving a Single Trigonometric Function
o based on “good” angles: i.e. sin() = 0.5, cos() = 1/2; tan() = 3
o based on inverse trigonometric functions: i.e. sin() = -.23, cos() = .432, tan() = 5
o always prefer EXACT numeric values of the solutions
o sometimes want only those solutions within the first rotation (i.e. 0 ≤  ≤ 2)
o sometimes want all real solutions to these equations
7.8: Solving Trigonometric Equations II
( text: 7, 23, 41, 53; others 5, 11, 13, 19, 21, 31, 33, 35)
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Quadratic Form i.e. 3sin2() + 5sin() ─ 2 = 0. Using Identities i.e. cos(2) = cos()
o often rewriting in terms of sine and cosine is an effective strategy; sometimes use factoring
o always prefer EXACT numeric values of the solutions
o sometimes want only those solutions within the first rotation (i.e. 0 ≤  ≤ 2)
o sometimes want all real solutions to these equations
8.1: Applications Involving Right Triangles
( text: 9, 19, 29, 39, 49, 51, 55, 63, 71; others: 15, 17, 21, 23, 25, 37, 41, 47, 57, 59, 69, 75, 79
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Finding Values of Trigonometric Functions Using Ratios of Sides in Right Triangles (soh cah toa)
Co- functions where co is a shortened form of the word complement
o A pair of angles are complementary if and only if their measures sum to a right angle measure
o Example: cosine literally translates to complement’s sine leading to cos( ) = sin(90─ )
Labeling: Vertices A, B, C; Side Lengths a (leg), b (leg), c (hypotenuse); Angles ,  , =90
Solving Right Triangles using Pythagorean Theorem: a2 + b2 = c2 and  +  = 90 (,  in degrees)
Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings
8.2: The Law of Sines
( text: 9, 23, 25, 31, 37, 39; others: 11, 19, 27, 29, 35, 41, 47, 53, 57)
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Triangles are Right (contain a right angle) OR Oblique (do not contain a right angle)
Labeling: Vertices A, B, C; Side Lengths a, b , c ; Angles ,  ,
Proving the Law of Sines
o Law of Sines (one form) For any triangle, sin α  sin β  sin γ
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o Law of Sines (another form) For any triangle,
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b
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sin α sin β sin γ
Solving Triangles: Apply the Law of Sines in two cases
o AAS two angles and a side (note as soon as we know two angles we know the third, why?)
o SSA two sides and an angle opposite one of these sides (this is the ambiguous case)
 Ambiguous case as possibilities include no triangle, one triangle or two triangles
Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings
8.3: The Law of Cosines
( text: 9, 23, 25, 31, 37, 39; others: 11, 19, 27, 29, 35, 41, 47, 53, 57)
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Proving the Law of Cosines
o Law of Cosines (one form)
For any triangle, a 2  b 2  c2  2bc cos
o Law of Cosines (another form)
For any triangle, b 2  a 2  c 2  2ac cos
o Law of Cosines (another form)
For any triangle, c 2  a 2  b 2  2ab cos
Double angle identities are a special case of the sum identities (for sine, cosine, tangent)
Solving Triangles Apply the Law of Cosines in two cases:
o SAS two sides and the included angle (the included angle is the one formed by these sides)
o SSS all three sides
Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings