Trigonometric Equations and Its Solutions

Trigonometric equations are those which involve trigonometric ratios. Finding the solution to a trigonometric equation requires finding the value of the angles satisfying the equation. If the interval of the solution is provided, the value of angles has to be found. If the interval is not given, the general solution has to be found.

What Is the System of Trigonometric Equations?

The mathematical expressions which include the ratios of the sides of a right-angled triangle can be called trigonometric expressions. An algebraic equation which involves trigonometric ratios and expressions is known as a trigonometric equation.

The solutions are principal solutions, if the variable of the equation lies in the interval 0 and 2π. It comprises all the solutions of trigonometric equations.

Important Results

The following formulae are used when solving a trigonometric equation:

Procedure to Find the General Solution of Trigonometric Equations

1] Deduce the given equation in the form of sin x, cos x, tan x.

2] Transform them into the below forms.

sin x = sin y

cos x = cos y

tan x = tan y

Use the first principal value of x as y. 

3] To find the general solutions of trigonometric equations, the following formulae are used:

sin x = sin y then x = nπ + (-1)ⁿy, where n is any integer

cos x = cos y then x = 2nπ + y or 2nπ – y, where n is any integer

tan x = tan y then x = nπ + y, where n is any integer

Also read

Trigonometric Equations

Solved Examples for IIT JEE

Example 1: If sin θ + cos θ = 1, then what is the general value of θ?

Solution:

sin θ + cos θ = 1 ⇒ (1 / √2) sin θ + (1 / √2) cos θ = 1 / √2

Dividing by √[1² + 1²] = √2,

We get, sin (θ + π / 4) = [1/ √2] = sin π / 4

⇒ θ + π / 4 = nπ + (−1)ⁿ [π / 4]

⇒ θ = nπ + (−1)ⁿ [π / 4] − π / 4

Example 2: If tan (cot x) = cot (tan x), what is sin 2x?

Solution:

Example 3: If the two angles on the base of a triangle are (22.5)^o and (112.5)^o, then what is the ratio of the height of the triangle to the length of the base?

Solution:

In Δ ACD, h / sin (67.5^o) = AC / sin 90^o

⇒ h / AC = sin (67.5^o) ….(i)

In ΔABC, AC / sin (22.5^o) = x / sin 45^o

⇒ AC / x = √2 / sin (22.5^o) …..(ii)

From (i) and (ii),

h / x = 1 / 2

Example 4: If the angles of a triangle be in the ratio 1 : 2 : 7, then what is the ratio of its greatest side to the least side?

Solution:

x + 2x + 7x = 180^o⇒ x = 18^o

Hence, the angles are 18^o, 36^o, 126^o

Greatest side = sin (126^o)

Smallest side = sin (18^o) and

Ratio = sin 126^o / sin (18^o) = [√5 + 1] / [√ 5 − 1]

Example 5: If the angles A, B, and C of a triangle are in A.P., and the sides a, b, and c opposite to these angles are in G. P., then a², b², c² are in which progression?

Solution:

Since A, B and C are in A.P., B = 60^∘ and b² = ac

cos B = a² + c²−b² / 2ac

⇒ 1 / 2 = a² + c²−b² / 2b² (∵ b² = ac)

b² = a² + c²−b²

⇒ a² + c² = 2b²

Example 6: Find the principal solutions of the below trigonometric equations.

a. sin x = 1/2

b. cos x  = -1/2

c. tan x = -1

Solution: The principal solution is in the range 0 ≤ x ≤ 2π. 

To find the value of the angle using the positive value,

The following identities need to be recalled to find the principal values:

a) sin x = 1/2

The principal solutions are π/6,  5π/6.

 b) cos x = -1/2

The principal solutions are 2π/3, 4π/3.

 c) tanx =-1

The principal solutions are 3π/4, 7π/4.

Example 7: Find the general solutions of the following.

Solution: 

Using the steps described above, 

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