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 . 5 10.5 Folium of Descartes . 14 3.10 Other Related Identities . 5 10.6 Spiral of Fermat . 14 3.11 Identities without Variables . 6 10.7 Cissoid of Diocles . 14 4 Graphs 6 10.8 Epispiral . 14 4.1 y = sin x .................... 6 10.9 Lituus . 15 4.2 y = cos x .................... 6 10.10 Eight Curve . 15 4.3 y = tan x .................... 6 10.11 Butterfly Curve . 15 4.4 y = csc x .................... 6 10.12 Strophoid . 15 4.5 y = sec x .................... 6 10.13 Cochleoid . 16 4.6 y = cot x .................... 6 10.14 Cycloid of Ceva . 16 10.15 Freeth’s Nephroid . 16 5 Tables 7 5.1 Exact Values of Trigonometric Functions . 7 11 Miscellaneous Stuff 17 5.2 Relations Between Trig Functions . 8 11.1 Pythagorean Triples . 17 11.2 Triangle Centers . 17 6 Inverse Trigonometric Functions 8 11.3 Area of the Triangle . 18 6.1 Principal Values . 8 11.4 Other Coordinate Systems . 18 1 Trigonometry Cram Sheet alltootechnical.tk 1 Definition 1.2 The Unit Circle y Triangle ABC has a right angle at C and sides of length a, b, c. The trigonometric functions of angle A are defined as (0, 1) √ √ follows: 1 3 1 3 − 2 , 2 2 , 2 √ √ √ √ 2 2 2 2 π − 2 , 2 2 , 2 a opposite 2 1. sin A = = 2π π √ 3 3 √ c hypotenuse 3 1 3 1 − , 3π ◦ π , 2 2 4 90 4 2 2 120◦ 60◦ 5π π b adjacent 6 6 2. cos A = = 150◦ 30◦ c hypotenuse (−1, 0) (1, 0) π 180◦ 3600◦◦ 2π x a opposite 3. tan A = = b adjacent 210◦ 330◦ 7π 11π 6 6 240◦ 300◦ √ 5π 7π √ c hypotenuse 3 1 270◦ 3 1 − 2 , − 2 4 4 2 , − 2 4. csc A = = 4π 5π a opposite 3 3 √ √ 3π √ √ 2 2 2 2 2 − 2 , − 2 2 , − 2 √ √ c hypotenuse − 1 , − 3 1 , − 3 5. sec A = = 2 2 2 2 b adjacent (0, −1) b adjacent 6. cot A = = a opposite 1.3 Degrees and Radians 1.1 Extensions to Angles > 90◦ A radian is that angle θ subtended at center O of a circle by an arc MN equal to the radius r. Since 2π radians = 360◦ A point P in the Cartesian plane has coordinates (x, y), we have: where x is considered as positive along OX and negative 0 0 along OX , while y is considered as positive along OY and 1 radian = 180◦/π = 57.29577951308232 ... ◦ negative along OY . The distance from origin O to point P is positive and denoted by r = px2 + y2. The angle A de- 1◦ = π/180 radians = 0.017453292519943 ... radians scribed counterclockwise from OX is considered positive. If it is described clockwise from OX it is considered negative. 1.4 Signs and Variations For an angle A in any quadrant, the trigonometric functions of A are defined as follows: Quadrant sin A cos A tan A + + + y I 1. sin A = (0, 1) (1, 0) (0, ∞) r + − − II (1, 0) (0, −1) (−∞, 0) x − − + 2. cos A = III r (0, −1) (−1, 0) (0, ∞) − + − y IV 3. tan A = (−1, 0) (0, 1) (−∞, 0) x Quadrant cot A sec A csc A r + + + 4. csc A = I y (∞, 0) (1, ∞) (∞, 1) − − + II r (0, −∞) (∞, −1) (1, ∞) 5. sec A = + − − x III (∞, 0) (−1, ∞) (∞, −1) x − + − 6. cot A = IV y (0, −∞) (∞, 1) (−1, ∞) 2 Trigonometry Cram Sheet alltootechnical.tk 2 Properties and General Forms 2.1.4 csc x Domain: {x|x 6= kπ, k ∈ }or S (kπ, (k + 1) π) 2.1 Properties Z k∈Z 2.1.1 sin x Range: {y|y ≤ 1 ∪ y ≥ 1} or (−∞, −1] ∪ [1, +∞) Period: 2π Domain: {x|x ∈ R} or (−∞, +∞) Range: {y| − 1 ≤ y ≤ 1} or [−1, 1] VA: x = kπ where k ∈ Z Period: 2π x-intercepts: none VA: none Parity: odd x-intercepts: kπ where k ∈ Z Parity: odd 2.1.5 sec x π S (k−1)π (k+1)π Domain: x|x 6= + kπ, k ∈ Z or , 2.1.2 cos x 2 k∈Z 2 2 Range: {y|y ≤ 1 ∪ y ≥ 1} or (−∞, −1] ∪ [1, +∞) Domain: {x|x ∈ R} or (−∞, +∞) Range: {y| − 1 ≤ y ≤ 1} or [−1, 1] Period: 2π π Period: 2π VA: x = 2 + kπ where k ∈ Z VA: none x-intercepts: none x-intercepts: π + kπ where k ∈ 2 Z Parity: even Parity: even 2.1.6 cot x 2.1.3 tan x S Domain: {x|x 6= kπ, k ∈ Z} or k∈ (kπ, (k + 1) π) Domain: x|x 6= π + kπ, k ∈ or S (k−1)π , (k+1)π Z 2 Z k∈Z 2 2 Range: {y|y ∈ R} or (−∞, +∞) Range: {y|y ∈ R} or (−∞, +∞) Period: π Period: π π VA: x = kπ where k ∈ Z VA: x = 2 + kπ where k ∈ Z x-intercepts: π + kπ where k ∈ x-intercepts: kπ where k ∈ Z 2 Z Parity: odd Parity: odd 2.2 General Forms of Trigonometric Functions Given some trigonometric function f (x), its general form is represented as y = Af (B (x − C)) + D, where its amplitude 2π π is |A|, its period is |B| or |B| (for tangent and cotangent), its phase shift is C, and its vertical translation is D units upward (if D > 0) or D units downward (if D < 0). The maximum and minimum value for sin x and cos x is A + D and −A + D respectively. amplitude period phase shift vertical translation 1 1 1 1 f f f f −1 0 1 2 3 −1 0 1 2 3 −1 0 1 2 3 −1 0 1 2 3 a −1 a −1 a −1 a −1 −2 −2 −2 −2 3 Trigonometry Cram Sheet alltootechnical.tk 3 Identities 3.2 Sum and Difference 3.1 Basic Identities sin (α ± β) = sin α cos β ± cos α sin β Reciprocal Identities cos (α ± β) = cos α cos β ∓ sin α sin β 1 1 tan α ± tan β csc θ = ; sin θ = tan (α ± β) = sin θ csc θ 1 ∓ tan α tan β 1 1 cot α cot β ∓ 1 sec θ = ; cos θ = cot (α ± β) = cos θ sec θ cot β ± cot α 1 1 cot θ = ; tan θ = tan θ cot θ 3.3 Double Angle sin θ csc θ = cos θ sec θ = tan θ cot θ = 1 sin 2α = 2 sin α cos α Ratio Identities cos 2α = cos2 α − sin2 α = 1 − 2 sin2 α = 2 cos2 α − 1 sin θ sin θ tan θ = ; cos θ = ; sin θ = cos θ tan θ 2 tan α cos θ tan θ tan 2α = 1 − tan2 α cos θ cos θ cot θ = ; sin θ = ; cos θ = sin θ cot θ sin θ cot θ 3.4 Half Angle Pythagorean Identities Let Qn, where n ∈ {1, 2, 3, 4}, denote the set of all angles within the nth quadrant of the Cartesian plane. sin2 θ + cos2 θ = 1; sin2 θ = 1 − cos2 θ; cos2 θ = 1 − sin2 θ 2 2 2 2 2 2 q 1−cos α α tan θ + 1 = sec θ; tan θ = sec θ − 1; sec θ − tan θ = 1 α 2 if 2 ∈ (Q1 ∪ Q2) sin = q 2 2 2 2 2 2 2 1−cos α α cot θ + 1 = csc θ; cot θ = csc θ − 1; csc θ − cot θ = 1 − 2 if 2 ∈ (Q3 ∪ Q4) q 1+cos α α Co-function Identities α 2 if 2 ∈ (Q1 ∪ Q4) cos = q π 2 − 1+cos α if α ∈ (Q ∪ Q ) sin − θ = cos θ 2 2 2 3 2 π α sin α 1 − cos α cos − θ = sin θ tan = = = csc α − cot α 2 2 1 + cos α sin α π tan − θ = cot θ 2 3.5 Multiple Angle π csc − θ = sec θ sin 3α = 3 sin α − 4 sin3 α 2 π 3 sec − θ = csc θ cos 3α = 4 cos α − 3 cos α 2 3 tan α − tan3 α π tan 3α = cot − θ = tan θ 2 2 1 − 3 tan α sin 4α = 4 sin α cos α − 8 sin3 α cos α Parity Identities cos 4α = 8 cos4 α − 8 cos2 α + 1 sin (−A) = − sin A 4 tan α − 4 tan3 α cos (−A) = cos A tan 4α = 1 − 6 tan2 α + tan4 α tan (−A) = − tan A n X n (n − i) π sin (nα) = cosi α sinn−i α sin csc (−A) = − csc A i 2 i=0 sec (−A) = sec A n X n (n − i) π cos (nα) = cosi α sinn−i α cos i 2 cot (−A) = − cot A i=0 4 Trigonometry Cram Sheet alltootechnical.tk 3.6 Power Reduction Definition 1 − cos 2θ sin2 θ = The two-argument form of the arctangent function, denoted 2 by tan−1 (y, x) gathers information on the signs of the in- puts in order to return the appropriate quadrant of the com- 1 + cos 2θ cos2 θ = puted angle.