Trigonometric Equations

Trigonometry involves the study of three sides. It is widely used to find and solve the sides and angles in the triangles, especially in the right angle triangle. In mathematics, various formulas of trigonometry are used to solve different types of trigonometric based problems. Although, it is also used for identifying unknown quantity angles with the help of other known quantity angles. Trigonometry consists of six different ratios. The values of these ratios are defined distinctively based on the sides of the right-angled triangle. Although, all the formulas of trigonometry are based on the Pythagoras theorem. 

Trigonometry 

Trigonometry is a wide field used to perform computation on the sides and angles of a right-angled triangle. There are various formulas of trigonometry which are used to perform such computations. Although, for defining these formulas of trigonometry, there are six trigonometric ratios, which are defined as follows:

• Cosine or cos y = hypotenuse/Perpendicular

• Sine or sin y =perpendicular/hypotenuse  

• Secant or sec y = hypotenuse/Base

• Cosecant or Cosec y =Base/hypotenuse  

• Cotangent or cot y = Base/Perpendicular

• Tangent or tan y = perpendicularhypotenuse

Here perpendicular is the side of the right-angle triangle, making the angle 90°. The base is also the side of the right-angled triangle on which the 90° angle is standing. The hypotenuse is the opposite side of the right angle in the given right-angled triangle. 

The general solution in trigonometry 

The functions which contain any trigonometric ratios or functions are called the trigonometric equations. In such equations, the solution is derived based on variables like x or y. Although, it depends on the equation of what alphabet of English it contains. The value of this variable lies within the range of  0 ≤ y ≤ 2π. This is called the principle solution of trigonometry. Although, if this principal solution contains integer ‘n’, it becomes the general solution m. 

Maths trigonometric formula 

Let’s learn about some maths trigonometric formulas. 

Trigonometry consists of a large number of formulas. These formulas are based on different domains. 

Basic formulas of trigonometry

There are three basic formulas of trigonometry. These are:

• sin2Y + cos2Y = 1

• sec2Y – tan2Y = 1

• cosec2Y – cot2Y = 1

Even or odd angle formulas of trigonometry. 

There are six even or odd angles formulas of trigonometry. These are:

• sin(-q) = -sin q

• cos(-q) = cos q

• tan(-q) = -tan q

• cot(-q) = -cot q

• sec(-q) = secq

• cosec(-q) = -cosec q

Complementary angles formulas of trigonometry. 

• sin (90° – ∠y) = cos y

• cos (90° – ∠y) = sin y

• tan (90° – ∠y) = cot y

• cot (90° – ∠y) = tan y

• sec (90° – ∠y) = cosec y

• cosec (90° – ∠y) = sec y.

Addition and subtraction formulas of trigonometry. 

Let’s learn about some addition and trigonometric subtraction formulas. 

• cos (p + q) = cos p cos q – sin p sin q

• cox (p – q) = cos p cos q + sin p sin q

• sin (p + q) = sin p cos q + cos p sin q

• sin (p – q) = sin p cos q – cos p sin q

• Tan (p + q) = tanp + tanq1-tanptanq 

• Tan(p – q) =tanp – tan q1 + tanptanq 

Some important formulas:

Let’s learn about some important formulas of trigonometry.

• 2 cos p cos q = cos (p+q) + cos (p-q)

• -2 sin p sin q = cos (p+q) – cos (p-q)

• 2 sin p cos q = sin (p+q) + sin (p-q)

• 2 cos p sin q = sin (p+q) – sin (p-q)

• cos(π-p) = -cos p

• cos(π+p) = -cos p

• sin(π-p) = sin p

• sin (π+p) = -sin p

• cos (2π-p) = cos p

• sin(2π-p) = – sin p

• cos (2 + p) = – sin p

• sin (2+ p) = cos x

Double angled based formulas of trigonometry. 

Let’s learn about the Double angled based formulas of trigonometry. 

• sin(2p) = 2sin(p) • cos(p) = 2tan p/1+tan2 p

• cos(2p) = cos2(p)–sin2(p) = 1-tan2 p/1+tan2 p

• cos(2p) = 2cos2(p)−1 = 1–2sin2(p)

• tan(2p) = 2tan(p)1−tan²(p) 

• sec (2p) = sec²p2 – sec²p

• cos (2p) = sec p. cos p2

Triple angled based formulas of trigonometry. 

Let’s learn about the triple angled based formulas of trigonometry. 

Sin 3p = 3sin p – 4sin3p

Cos 3p = 4cos3p-3cos p

Tan 3p = 3tanp-tan³p1-3tan²p

Conclusion 

Trigonometry is a broad field that consists of various domains. However, trigonometry is a triangle-based study, but it still covers the various mathematics domains. However, trigonometry can be used to define other derivatives and formulas of mathematics. It is also used in astronomy and physics widely. Formulas of trigonometry typically deal with the three sides of the right-angle triangle, which are hypotenuse, base and perpendicular. All the formulas of trigonometry are derived from the Pythagoras theorem of right-angled triangles. The six trigonometry ratios are derived from the Pythagoras theorem by taking one side or angle as a main.