Trigonometric Equations And Its Solutions

Trigonometry was an offshoot of geometry in its early stages; the two fields of mathematics did not become independent until the 16th century. Trigonometry is a discipline of mathematics that investigates triangles’ sides and angles. Every straight-sided form may be broken into a series of triangles. Hence trigonometric equations are present throughout the geometry. Furthermore, trigonometry includes a bewildering number of interconnections with other fields of mathematics, including complex numbers, infinite series, and logarithms. In-plane trigonometry deals with problems involving angles and lengths in a single plane. Spherical trigonometric equations consider applications to similar issues in more than one plane of three-dimensional space.

Formulas for Basic Trigonometric Functions:

In Trigonometry, there are essentially six ratios used to find the elements. Trigonometric functions are what they’re called.Secant, cosecant, sine, cosine, tangent, and cotangent are the six trigonometric equation formulas.

The trigonometric calculations and identities are derived using a right-angled triangle:

• Sin = Opposite SideHypotenuse

• cos = Adjacent SideHypotenuse

•  tan = Opposite SideAdjacent Side

• sec = HypotenuseAdjacent Side

• cosec = HypotenuseOpposite Side

• cot = Adjacent SideOpposite Side

Types of trigonometric Equation solutions include:-

The types of trigonometric equations are as follows in mathematics:

Principal Solution: For trigonometric functions, the initial values of angles are referred to as principal solutions. The Sin x and Cos x solutions repeat after a 2-radian interval, while the Tan x solution repeats after a radian interval. Principal solutions answer all trigonometric equations in which x is between 0 and 2.

The solution in general: If the angles are the same for the same trigonometric function response, the general solution of the trigonometric function is found. All trigonometric equation answers beyond 2 are collected and given as a generic trigonometric calculation solution.

The following are the general solutions for Sin, Cos, and Tan.

Sin = Sin, and the general solution is = n + (-1) n, where

 n Z Cos = Cos, and the general solution is = 2n +, where 

n Z Tan = Tan and the general solution is = n.

Example:

Students can use the solved problems below to connect the trigonometric equation formulas.

1). Find x given f(x) = g(x), g(x) = cot (x – 50°), and h(x) = cos x in Example 1. (x). 

Also, get cosec x + tan3x if h(x) = 45

Solution: If f(x) = g(x), then tan 3x = cot (x –50°), cot (90° – 3x) = cot (x-50°), 90° – 3x = x – 50, or x = 35.

We have cos x = 45 for h(x)=cos x and h(x) = 45

As a result, sin x equals 35, cosec x equals 53, and tan x equals 45

Alternatively, 

cosec x + tan 3x = (53) + (45)3 

= 817375= 2.178

2) Find all sin2(E2) cos2(E2) = 1 solutions.

Solution: To get the equation in terms of one function of one angle, you apply identities in step one. You have a choice in which identity you wish to utilise, as is common in trig. You may apply the double-angle formula cos(2u) = cos2(u) sin2(u) with u = E2, or replace cos2(E2) with sin2(E2). The second method yields the following equation:

cos2(E2) = 1 sin2(E2)

sin2(E2) = 1 cos2(E2)

2E2= cos(2E2) = 1

cos(E) = 1 

Step 2: It has already been completed.

Step 3: Make a note of the angle:

E = + 2n = (2n+1)

Because the angle is the variable, step 4 is already completed.

Step 5: It focuses on trigonometric calculation straightforward: 

the problem specifically requests all E values.

For any integer n, the answer is E =(2n+1).

3). Prove that Cos x = cos y implies x = 2n y, where n = Z, for any two real integers x and y.

Proof: If sin x = sin y, sin x – sin y = 0

-2 sin x+y2 sin x+y2 sin x+y2 = 0 [Using the formula – 2 sin 12 (A + B) sin 12 (A – B)]

0 = sin x+y2 or 0 = sin x+y2

x+y2 = n 

(x y)2 = n

(x y)2 [Sin A = 0 implies A = n, where n equals Z]

x = 2n – y or x = 2n + y, where n is greater than Z.

As a result, x = 2n y, where n = Z.

Conclusion:

Steps to Solve a Trigonometric Equation formula: There are several methods for solving trigonometric calculations. Convert a trigonometric equation to a single trigonometric ratio equation (sin, cos, Tan). Convert a trigonometric equation’s numerous angles, or submultiple angles, into a single angle. Make a polynomial, quadratic, or linear equation out of the equation. Find the value of the trigonometric ratio by solving the trigonometric equation similarly to normal equations. The solution of the trigonometric equation is represented by the angle of the trigonometric ratio or its value. Trigonometry is rich in trigonometry equation formulas.