Trigonometry Formula

Trigonometry Formula

Jul 20, 2023, 16:45 IST

In mathematics, trigonometry is one of the most important topics to learn. Trigonometry is the study of triangles, where "Trigon" means triangle and "metry" means measurement.

A list of trigonometric formulas was formulated for the right triangle. All trigonometric formulas are based on identities and ratios. The relationship between a triangle's side and angle lengths is formulated using trigonometric concepts.

A list of formulas based on trigonometry will help students solve trigonometry problems quickly. Below is a list of formulas based on the right triangle and the unit circle that can be used for studying trigonometry.

List of Important Trigonometry Formulas

Let us learn trigonometry formulas, considering an right-angled triangle with an angle θ, a hypotenuse, a side adjacent to angle θ, and a side opposite the angle to angle θ.

Trigonometric Ratios

• sinθ = Opposite side / Hypotenuse
• cosθ = Adjacent side / Hypotenuse
• tanθ = Opposite side / Adjacent Side
• secθ = Hypotenuse / Adjacent side
• cosecθ = Hypotenuse / Opposite side
• cotθ = Adjacent Side / Opposite side

Trigonometric Ratios for Unit Circle

Generally, for a unit circle, the radius equals 1, and θ is the angle. The value of the hypotenuse and adjacent side here is equal to the radius of the unit circle.

Hypotenuse = Adjacent side to θ = 1

Therefore, the ratios of trigonometry are given by:

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

Trigonometry Identities

Tangent and Cotangent Identities

• tanθ = sin θ / cos θ
• cotθ = cos θ / sin θ

Reciprocal Identities

• sinθ = 1/cosecθ
• cosecθ = 1/sinθ
• cosθ = 1/secθ
• secθ = 1/cosθ
• tanθ = 1/cotθ
• cotθ = 1/tanθ

Pythagorean Identities

• sin2θ + cos2θ = 1
• 1 + tan2θ = sec2θ
• 1 + cot2θ = cosec2θ

Even and Odd Angle Formulas

• sin(-θ) = -sinθ
• cos(-θ) = cosθ
• tan(-θ) = -tanθ
• cot(-θ) = -cotθ
• sec(-θ) = secθ
• cosec(-θ) = -cosecθ

Co-function Formulas

• sin(900-θ) = cosθ
• cos(900-θ) = sinθ
• tan(900-θ) = cotθ
• cot(900-θ) = tanθ
• sec(900-θ) = cosecθ
• cosec(900-θ) = secθ

Double Angle Formulas

• sin2θ = 2 sinθ cosθ
• cos2θ = 1 – 2sin2θ
• tan2θ = 2 tanθ / 1-tan2θ

Half Angle Formulas

• sinθ  =  ±√1-cos2θ/2
• cosθ = ±√1+cos2θ/2
• tanθ = ±√1+cos2θ/1-cos2θ

Thrice of Angle Formulas

• sin3θ = 3sinθ – 4 sin3θ
• Cos 3θ = 4cos3θ – 3 cosθ
• Tan 3θ = 3tanθ - tan3θ/ 1- 3tan2θ
• Cot 3θ = cot3θ - 3cotθ/3cot2θ - 1

Product to Sum Formulas

• Sin A Sin B = 1/2 [Cos (A-B) – Cos (A+B)]
• Cos A Cos B = 1/2 [Cos (A-B) + Cos (A+B)]
• Sin A Cos B = 1/2 [Sin (A+B) + Sin (A-B)]
• Cos A Sin B = 1/2 [Sin (A+B) – Sin (A-B)]

Sum to Product Formulas

• Sin A + Sin B = 2 sin (A+B)/2 cos (A-B)/2
• Sin A - Sin B = 2 sin (A+B)/2 sin (A-B)/2
• Cos A + Cos B = 2 cos (A+B)/2 cos (A-B)/2
• Cos A - Cos B = 2 cos (A+B)/2 cos (A-B)/2

Inverse Trigonometric Functions

If Sin θ = x, then θ = sin-1 x = arcsin(x)

Similarly,

θ = cos-1x = arccos(x)

θ = tan-1 x = arctan(x)

Also, the inverse properties could be defined as;

sin-1(sin θ) = θ

cos-1(cos θ) = θ

tan-1(tan θ) = θ

Trigonometry Table

Frequently Asked Question (FAQs)

Q1. How many total formulas are in trigonometry?

Ans. The six trigonometric functions are sine, secant, cosine, cosecant, tangent, and cotangent. By using a right-angled triangle as a reference, the trigonometric functions and identities are derived: sin θ = Opposite Side/Hypotenuse. cos θ = Adjacent Side/Hypotenuse.

Q2. What are the 4 types of trigonometry?

Ans. There are 4 types of trigonometry used today, which include core, plane,  spherical and analytic. Core trigonometry deals with the ratio between the angles of a right triangle and its sides.

Q3. What is a limit of trigonometric function?

Ans. Consider the sine function f(x)=sin(x), where x is measured in radian. The sine function is continuous everywhere, as we see in the graph above:, therefore, limx→csin(x)=sin(c).

Q4. What is the range of Cos function?

Ans. −1 ≤ cosx ≤ 1

Q5. Where does sin equal 0?

Ans. sin x = 0 is for x = nπ where n is an integer e.g., -3,-2,-1,0,1,2,3,......... and x is in radian.

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