Trigonometrical Equations	Solutions
sin θ = 0	θ = nπ
cos θ = 0	θ = (nπ + π/2)
tan θ = 0	θ = nπ
sin θ = 1	θ = (2nπ + π/2) = (4n+1) π/2
cos θ = 1	θ = 2nπ
tan θ = 1	θ = nπ + π/4
sin θ = sin α	θ = nπ + (-1)n α, where α ∈ [-π/2, π/2]
cos θ = cos α	θ = 2nπ ± α, where α ∈ (0, π]
tan θ = tan α	θ = nπ + α, where α ∈ (-π/2, π/2]
sin 2θ = sin 2α	θ = nπ ± α
cos 2θ = cos 2α	θ = nπ ± α
tan 2θ = tan 2α	θ = nπ ± α

​Trigonometric Equations: Practice Questions with Solution

Question 1: Solve for θ if sin (θ) = 1/2

Solution:

To solve for θ given that

sin(θ) = 1/2​

We need to find the angles where the sine function equals 1/2.

Sine function equals 1/2 at specific angles within the standard range of o to 360 degree (or 0 to 2π radians):

sin(θ) = 1/2

θ = 30^o,150^o (in degrees)

θ = π/6, 5π/6 ( in radians)

Question 2: Solve for θ if cos (θ) = -1/2.

Solution:

To solve for θ given that

cos(θ) = -1/2​,

We need to find the angles where the sine function equals -1/2.

θ = arccos (-1/2) = 120^o or 240^o

θ = 120^o + 360^o(n) or 240^o + 360^o(n), where n = 1, 2……

Question 3: Solve for θ if tan(θ) = 1

Solution:

To solve for θ given that

tan(θ) = 1​,

we need to find the angles where the sine function equals 1.

θ = arctan (1) = 45^o or 225^o

θ = 45^o + 360^o(n), where n = 1, 2……

​Question 4: Solve for θ: 2cos²θ – 1 = 0.

Solution:

2cos²θ – 1 = 0

cos²θ = 1/2

cosθ = ± 1/√2

θ = 45^o, 135^o, 225^o, 315^o

Question 5: Solve 2sin(θ) + 1 = 0 for 0 ≤ θ ≤ 2π.

Solution:

2sin(θ) + 1 = 0

sin(θ) = -1/2

θ = 7π/6, 11π/6

Question 6: Solve sin(θ) cos(θ) = 1/4 for 0 ≤ θ ≤ 2π.

Solution:

First, use the double-angle identity for sine:

sin(θ) cos(θ) = (1/2) sin(2θ)

Thus, the equation becomes:(1/2) sin(2θ) = 1/4

Multiply both sides by 2 to solve for sin(2θ):

sin(2θ) = 1/2

Sine function has a value of 1/2 at the angles:

2θ = π/6, 5π/6, 13π/6, 17π/6

Next, solve for θ:

θ = π/12, 5π/12, 13π/12, 17π/12

Thus, the solution to the equation sin(θ) cos (θ) = 1/4 for 0 ≤ θ ≤ 2 are:

θ = π/12, 5π/12, 13π/12, 17π/12

Question 7: Solve tan(2θ) = √3 for 0 ≤ θ ≤ 2π

Solution:

​First, solve for 2θ in the equation tan(2θ) = √3 .

Tangent function has a value of √3 at the angles: .

2θ = π/3, 4π/3

Next, solve for θ:

θ = π/6, 2π/3, 7π/6, 5π/3

Thus, the solutions to the equation tan(2θ) = √3 for 0 ≤ θ ≤ 2π are:

θ = π/6, 2π,/3, 7π/6, 5π/3

Question 8: Solve cos(3θ) = 1 for 0 ≤ θ < 2π.

Solution:

First, solve for 3θ in the equation cos(3θ) = 1.

Cosine function has a value of 1 at the angles:

3θ = 0, 2π, 4π, 6π

Next, solve for θ:θ = 0, 2π/3 , 4π/3

Thus, the solution to the equation cos(3θ) = 1 for 0 ≤ θ ≤ 2π are:θ = 0, 2π/3, 4π/3

Trigonometric Equations: Worksheet

The worksheet on trigonometric equation is added in form of image below:

Answer Key

• Ans 1. θ = 4π/3, 5π/3
• Ans 2. θ = π/4, 3π/4, 5π/4, π
• Ans 3. θ = π/3, 4π/3
• Ans 4. θ = 0, π/3, 2π/3, π, 4π/3
• Ans 5. θ = π/4, 7π/4
• Ans 6. θ = π/3, 2π/3, 4π/3, π
• Ans 7. θ = π/6, 5π/6
• Ans 8. θ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4
• Ans 9. θ = π/6, 7π/6, 11π/6
• Ans 10. θ = 3π/8, 7π/8, 11π/8, 15π/8

Conclusion

Trigonometric equations are a critical component of mathematics, providing essential tools for solving problems in various scientific and engineering disciplines. By mastering these equations, students can enhance their analytical skills and prepare for advanced studies.

Related Articles:

• Trigonometric Identities Practice Problems
• Important Trigonometry Questions
• Trigonometry in Maths

Frequently Asked Questions

What are Trigonometric Equations?

Trigonometric functions, such as sine, cosine, and tangent, are used in trigonometric equations in mathematics. They are employed to identify angles that meet predetermined criteria.

What is Significance of Trigonometric Equations?

In many disciplines, including physics, engineering, and astronomy, trigonometric equations are crucial because they facilitate the analysis of periodic occurrences and the resolution of practical issues.

What are Basic Trigonometric Functions?

The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

How are Trigonometric Equations Solved?

Numerous techniques, including factoring, using trigonometric identities, graphical methods, and inverse trigonometric functions, can be used to solve trigonometric equations.

What are Pythagorean identities?

The sine, cosine, and tangent squares of functions are related by the basic trigonometric identities known as Pythagorean identities.

How are Trigonometric Equations Used in Real Life?

Trigonometric equations are used in various applications such as signal processing, sound waves analysis, electrical engineering, and navigation.