TRIGONOMETRY

Trigonometry is the relationship between the sides and angles of a right-angled triangle.

Following are the size basic trigonometric function :

• sin = Opposite side/ Hypotenuse
• cos = Adjacent side / Hypotenuse
• tan = Opposite side / Adjacent side
• cosec = Hypotenuse / Opposite side
• sec = Hypotenuse / Adjacent side
• cot = Adjacent side / Opposite side

Graphs of Basic Trigonometry function :

Following are the domain and range of each function :

• sin : R → [– 1, 1]
• cos : R → [– 1, 1]
• tan : R – { x : x = (2n + 1) π/2, n ∈ Z} →R
• cot : R – { x : x = nπ, n ∈ Z} →R
• sec : R – { x : x = (2n + 1) π/2, n ∈ Z} →R – (– 1, 1)
• csc : R – { x : x = nπ, n ∈ Z} →R – (– 1, 1)

1.) sin function:

2.) cos function:

3.) tan function:

Trigonometry Angles :

The trigonometry angles commonly used in trigonometry problems are 0°, 30°, 45°, 60° and 90°.

Basic Trigonometric indenties :

• sin²θ + cos²θ = 1
• tan²θ + 1 = sec²θ
• cot²θ + 1 = cosec²θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ – sin²θ
• tan 2θ = 2 tan θ / (1 – tan²θ)
• cot 2θ = (cot²θ – 1) / 2 cot θ

Reciprocal Identities :

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

Sine and Cosine Laws :

If A, B and C are angles and a, b and c are the sides of a triangle, then,

• a/sinA = b/sinB = c/sinC
• c² = a² + b² – 2ab • cos C
• a² = b² + c² – 2bc • cos A
• b² = a² + c² – 2ac • cos B

Basic Trigonometric sum and difference indenties :

For angles u and v, we have the following relationships :

• sin(u + v) = sin(u) • cos(v) + cos(u) • sin(v)
• cos(u + v) = cos(u) • cos(v) – sin(u) • sin(v)
• tan(u+v) = (tan(u) + tan(v))/(1−tan(u)) • tan(v)
• sin(u – v) = sin(u) • cos(v) – cos(u) • sin(v)
• cos(u – v) = cos(u) • cos(v) + sin(u) • sin(v)
• tan(u-v) = (tan(u) − tan(v))/(1+tan(u)) • tan(v)

Periodicity Identities (in Radians) :

These formulas are used to shift the angles by π/2, π, 2π, etc. :

• sin (π/2 – A) = cos A , cos (π/2 – A) = sin A)
• sin (π/2 + A) = cos A , cos (π/2 + A) = – sin A
• sin (3π/2 – A) = – cos A , cos (3π/2 – A) = – sin A
• sin (3π/2 + A) = – cos A , cos (3π/2 + A) = sin A
• sin (π – A) = sin A , cos (π – A) = – cos A
• sin (π + A) = – sin A , cos (π + A) = – cos A
• sin (2π – A) = – sin A , cos (2π – A) = cos A
• sin (2π + A) = sin A , cos (2π + A) = cos A

Periodicity Identities (in Degrees) :

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = csc x
• csc(90°−x) = sec x

Sign of Trigonometric Functions :

• sin(-θ) = -sin θ
• cos(-θ) = cos θ
• tan(-θ) = -tan θ
• cot(-θ) = -cot θ
• sec(-θ) = sec θ
• cosec(-θ) = -cosec θ

Double Angle Identities :

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan² x)]
• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan² x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin²(x)
• tan(2x) = [2tan(x)]/ [1−tan²(x)]
• sec (2x) = sec2 x/(2-sec2 x)
• csc (2x) = (sec x. csc x)/2

Triple Angle Identities :

• sin 3x = 3sin x – 4sin3x
• cos 3x = 4cos3x-3cos x
• tan 3x = [3tanx-tan^3x]/[1-3tan²x]
• cot 3x = [cot^3x-3cotx]/[3cot²x-1]

Half Angle Identities :

Product identities :

• sinx • cosy = (sin(x+y)+sin(x−y))/2
• cosx • cosy = (cos(x+y)+cos(x−y))/2
• sinx • siny = (cos(x-y)-cos(x+y))/2

Sum to Product identities :

• sinx + siny = 2sin((x+y)/2) • cos((x-y)/2)
• sinx - siny = 2cos((x+y)/2) • sin((x-y)/2)
• cosx + cosy = 2cos((x+y)/2) • cos((x-y)/2)
• cosx - cosy = -2sin((x+y)/2) • sin((x-y)/2)

Inverse Trigonometry Formulas :

• sin^-1 (1/a) = cosec^-1(a) , a ≥ 1 or a ≤ – 1
• cos^-1(1/a) = sec^-1(a) , a ≥ 1 or a ≤ – 1
• tan^-1(1/a) = cot^-1(a) , a>0
• sin^-1(–a) = – sin^-1(a) , a ∈ [– 1, 1]
• tan^-1(–a) = – tan^-1(a) , a ∈ R
• cosec^-1(–a) = –cosec^-1(a) , | a | ≥ 1
• cos^-1(–a) = π – cos^-1(a) , a ∈ [– 1, 1]
• sec^-1(–a) = π – sec^-1(a) , | a | ≥ 1
• cot^-1(–a) = π – cot^-1(a) , a ∈ R

Addition Properties :

• sin^-1a + cos^-1a = π/2 , a ∈ [– 1, 1]
• tan^-1a + cot^-1a = π/2 , a ∈ R
• cosec^-1a + sec^-1a = π/2 , | a | ≥ 1
• tan^-1a + tan^1 b = tan^-1 [(a+b)/1-ab] , ab<1< /li>
• tan^-1a – tan^-1 b = tan^-1 [(a-b)/1+ab] , ab>-1
• tan^-1a – tan^-1 b = π + tan^-1[(a+b)/1-ab] , ab > 1 ; a,b > 0

Twice of Inverse of tan Function :

• 2tan^-1a = sin^-1 [2a/(1+a²)] , |a| ≤ 1
• 2tan^-1a = cos^-1[(1-a²)/(1+a²)] , a ≥ 0
• 2tan^-1a = tan^-1[2a/(1+a²)] , – 1 < a < 1

Inverse Trigonometry Substitution :

Complementary Ratios :

Quadrant I :

• sin(π/2−θ) = cos θ
• cos(π/2−θ) = sin θ
• tan(π/2−θ) = cot θ
• cot(π/2−θ) = tan θ
• sec(π/2−θ) = cosec θ
• cosec(π/2−θ) = sec θ

Quadrant II :

• sin(π−θ) = cos θ
• cos(π−θ) = -cos θ
• tan(π−θ) = -tan θ
• cot(π−θ) = – cot θ
• ec(π−θ) = -sec θ
• cosec(π−θ) = cosec θ

Quadrant III :

• sin(π+ θ) = – sin θ
• cos(π+ θ) = – cos θ
• tan(π+ θ) = tan θ
• cot(π+ θ) = cot θ
• sec(π+ θ) = -sec θ
• cosec(π+ θ) = -cosec θ

Quadrant IV :

• sin(2π− θ) = – sin θ
• cos(2π− θ) = cos θ
• tan(2π− θ) = – tan θ
• cot(2π− θ) = – cot θ
• sec(2π− θ) = sec θ
• cosec(2π− θ) = -cosec θ