Trigonometry Cheat Sheet

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Trigonometry Formulas and Properties Reciprocal Identities: Tangent and Cotangent Identities: sin cos 1 1 tan = cot = sin = csc = cos sin csc sin 1 1 cos = sec = sec cos Pythagorean Identities: sin + cos = 1 1 1 tan = cot = tan2 + 1 =2sec cot tan 2 2 1 + cot = csc 2 2 Even/Odd Formulas: Cofunction Formulas: sin( ) = sin cos( ) = cos tan( ) = tan sin = cos csc = sec tan = cot 2 2 2 csc(−) = − csc sec(−) = sec cot(−) = − cot cos� − � = sin sec� − � = csc cot � − � = tan 2 2 2 − − − − − � − � � − � � − � Product to Sum Formulas: Sum to Product Formulas: 1 + sin sin = [cos( ) cos( + )] sin + sin = 2 sin cos 2 2 2 1 + − cos cos = [cos( − )−+ cos( + )] sin sin = 2 cos� � sin � � 2 2 2 1 + − sin cos = [sin( +− ) + sin( )] cos +−cos = 2 cos� � cos� � 2 2 2 1 + − cos sin = [sin( + ) sin( − )] � � � � 2 cos cos = 2 sin sin 2 2 − − − − − � � � � Sum and Difference Formulas: Half-Angle Formulas: Double Angle Formulas: ( ) sin( ± ) = sin cos ± sin cos 1 cos sin 2 = 2 sin cos sin = ± 2 2 cos( ± ) = cos cos sin sin − � � � cos(2 ) = cos sin tan ± tan 1 + cos tan( ± ) = ∓ cos = ± 2 2 1 tan tan 2 2 = 2 cos 1− 2 � = 1 2 sin ∓ � � − Periodic Formulas: 1 cos 2 tan = ± − sin( + 2 ) = sin csc( + 2 ) = csc 2 1 + cos 2tan − tan 2 = � � � 1 tan cos( + 2) = cos sec( + 2) = sec 2 − tan( + ) = tan cot( + ) = cot Updated: October 2019 Trigonometric Functions: Right Triangle: Unit Circle: y hypotenuse ( , ) opposite r x θ adjacent opposite hypotnuse sin = csc = hypotnuse opposite sin = csc = θ θ adjacent hypotnuse cos = sec = hypotnuse adjacent cos = sec = θ θ opposite adjacent tan = cot = adjacent opposite tan = cot = θ θ Inverse Trigonometric Functions: Definition: Alternative Definition: = sin = sin sin =arcsin −1 −1 = cos �same��as = cos cos =arccos −1 −1 �same��as tan =arctan = tan = tan Law of Sines: = = −1 sin sin sin −1 �� Domain and Range: Law of Cosines: = + 2 cos Function Domain Range 2 2 2 = sin 1 1 − 2 2 = + 2 cos −1 − ≤ ≤ − ≤ ≤ 2 2 2 = cos 1 1 0 = + −2 cos −1 − ≤ ≤ ≤ ≤ = tan < < 2 2 2 2 − −1 −∞ ≤ ≤ ∞ − Law of Tangents: = cot 0 1 tan ( ) −1 = 2 −∞ ≤ ≤ ∞ ≤ ≤ + 1 = sec 1, 1 0 < , < tan (+ −) 2 2 − 2 −1 ≤ − ≥ ≤ ≤ 1 = csc 1, 1 < 0 , 0 < tan ( ) 2 2 = 2 −1 1 + tan ( + ) ≤ − ≥ − ≤ ≤ − 2 − 1 tan ( ) Inverse Properties: = 2 1 + tan ( + ) − 2 − sin (sin ( )) = sin (sin( )) = −1 −1 cos (cos ( )) = cos (cos( )) = −1 −1 tan (tan ()) = tan (tan()) = −1 −1 Updated: October 2019 .

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Trigonometry Cram Sheet
Trigonometry Cram Sheet August 3, 2016 Contents 6.2 Identities . 8 1 Deﬁnition 2 7 Relationships Between Sides and Angles 9 1.1 Extensions to Angles > 90◦ . 2 7.1 Law of Sines . 9 1.2 The Unit Circle . 2 7.2 Law of Cosines . 9 1.3 Degrees and Radians . 2 7.3 Law of Tangents . 9 1.4 Signs and Variations . 2 7.4 Law of Cotangents . 9 7.5 Mollweide’s Formula . 9 2 Properties and General Forms 3 7.6 Stewart’s Theorem . 9 2.1 Properties . 3 7.7 Angles in Terms of Sides . 9 2.1.1 sin x ................... 3 2.1.2 cos x ................... 3 8 Solving Triangles 10 2.1.3 tan x ................... 3 8.1 AAS/ASA Triangle . 10 2.1.4 csc x ................... 3 8.2 SAS Triangle . 10 2.1.5 sec x ................... 3 8.3 SSS Triangle . 10 2.1.6 cot x ................... 3 8.4 SSA Triangle . 11 2.2 General Forms of Trigonometric Functions . 3 8.5 Right Triangle . 11 3 Identities 4 9 Polar Coordinates 11 3.1 Basic Identities . 4 9.1 Properties . 11 3.2 Sum and Diﬀerence . 4 9.2 Coordinate Transformation . 11 3.3 Double Angle . 4 3.4 Half Angle . 4 10 Special Polar Graphs 11 3.5 Multiple Angle . 4 10.1 Limaçon of Pascal . 12 3.6 Power Reduction . 5 10.2 Rose . 13 3.7 Product to Sum . 5 10.3 Spiral of Archimedes . 13 3.8 Sum to Product . 5 10.4 Lemniscate of Bernoulli . 13 3.9 Linear Combinations .
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Chapter 6 Additional Topics in Trigonometry
1111572836_0600.qxd 9/29/10 1:43 PM Page 403 Additional Topics 6 in Trigonometry 6.1 Law of Sines 6.2 Law of Cosines 6.3 Vectors in the Plane 6.4 Vectors and Dot Products 6.5 Trigonometric Form of a Complex Number Section 6.3, Example 10 Direction of an Airplane Andresr 2010/used under license from Shutterstock.com 403 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1111572836_0601.qxd 10/12/10 4:10 PM Page 404 404 Chapter 6 Additional Topics in Trigonometry 6.1 Law of Sines Introduction What you should learn ● Use the Law of Sines to solve In Chapter 4, you looked at techniques for solving right triangles. In this section and the oblique triangles (AAS or ASA). next, you will solve oblique triangles—triangles that have no right angles. As standard ● Use the Law of Sines to solve notation, the angles of a triangle are labeled oblique triangles (SSA). A, , and CB ● Find areas of oblique triangles and use the Law of Sines to and their opposite sides are labeled model and solve real-life a, , and cb problems. as shown in Figure 6.1. Why you should learn it You can use the Law of Sines to solve C real-life problems involving oblique triangles.
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The Proofs of the Law of Tangents
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Trigonometry in the AIME and USA(J)MO
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Notes from Trigonometry
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A History of Trigonometry Education in the United States: 1776-1900
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