Solving Equations Involving a Single Trigonometric Function
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)
· Solving Equations Involving a Single Trigonometric Function
o based on “good” angles: i.e. sin(q) = 0.5, cos(q) = 1/Ö2; tan(q) = Ö3
o based on inverse trigonometric functions: i.e. sin(q) = -.23, cos(q) = .432, tan(q) = Ö5
o always prefer EXACT numeric values of the solutions
o sometimes want only those solutions within the first rotation (i.e. 0 ≤ q ≤ 2p)
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)
· Quadratic Form i.e. 3sin2(q) + 5sin(q) ─ 2 = 0. Using Identities i.e. cos(2q) = cos(q)
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 ≤ q ≤ 2p)
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
· 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(q ) = sin(90°─ q)
· Labeling: Vertices A, B, C; Side Lengths a (leg), b (leg), c (hypotenuse); Angles a, b , g =90°
· Solving Right Triangles using Pythagorean Theorem: a2 + b2 = c2 and a + b = 90° (a, b 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)
· 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 a, b , g
· Proving the Law of Sines
o Law of Sines (one form) For any triangle,
o Law of Sines (another form) For any triangle,
· 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)
· Proving the Law of Cosines
o Law of Cosines (one form) For any triangle,
o Law of Cosines (another form) For any triangle,
o Law of Cosines (another form) For any triangle,
· 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