Trigonometric Formulas

Trigonometry is the study of relationships between angles, lengths, and heights of triangles. It includes ratios, function, identities, formulas to solve problems based on it, especially for right-angled triangles. Now It is a completely new and tricky chapter where one needs to learn all the formulas and apply them accordingly. So we decide to compile them all together so that it becomes easy for you to revise the formulas on the go.

The trigonometric formulas for ratios are majorly based on the three sides of a right-angled triangle, such as the adjacent side or base, perpendicular and hypotenuse (See the above figure). Applying Pythagoras theorem for the given right-angled triangle, we have:

(Perpendicular)2+(Base)2=(Hypotenuse)2

⇒(P)2+(B)2=(H)2

Now, let us see the formulas based on trigonometric ratios (sine, cosine, tangent, secant, cosecant and cotangent)

Basic Trigonometric formulas

The Trigonometric formulas are given below:

Reciprocal Relation Between Trigonometric Ratios

Trigonometric Sign Functions

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

Trigonometric Identities

• sin2A + cos2A = 1
• tan2A + 1 = sec2A
• cot2A + 1 = cosec2A

Periodic Identities

• sin(2nπ + θ ) = sin θ
• cos(2nπ + θ ) = cos θ
• tan(2nπ + θ ) = tan θ
• cot(2nπ + θ ) = cot θ
• sec(2nπ + θ ) = sec θ
• cosec(2nπ + θ ) = cosec θ

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) = 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 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

Sum and Difference of Two Angles

• sin (A + B) = sin A cos B + cos A sin B
• sin (A − B) = sin A cos B — cos A sin B
• cos (A + B) = cos A cos B — sin A sin B
• cos (A — B) = cos A cos B + sin A sin B
• tan(A+B) = [(tan A + tan B)/(1 — tan A tan B)]
• tan(A-B) = [(tan A — tan B)/(1 + tan A tan B)]

Double Angle Formulas

• sin2A = 2sinA cosA = [2tan A + (1+tan2A)]
• cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)]
• tan 2A = (2 tan A)/(1-tan2A)

Triple Angle Formulas

• sin3A = 3sinA — 4sin3A
• cos3A = 4cos3A — 3cosA
• tan3A = [3tanA–tan3A]/[1−3tan2A]

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Now that we’re done with the formulas, you know the drill. Yes! It’s time for practice and preparation for your exams. Now while solving questions you might have a doubt, in that case, we’ve got you covered as well, just click here and get a 1v1 session with Subject Experts and get your doubt solved!