Problems on Trigonometric Equation

We will learn how to solve different types of problems on trigonometric equation containing one or many trig functions. First we need to solve the trigonometric function (if required) and then solve for the angle value using the trigonometric equation formulas.

1. Solve the equation sec θ - csc θ = 4/3

Solution:

sec θ - csc θ = 4/3

⇒ \(\frac{1}{cos θ}\) - \(\frac{1}{sin θ}\) = 4/3

⇒ \(\frac{sin θ - cos θ}{sin θ  cos θ}\) = 4/3

⇒ 3 (sin θ - cos θ) = 4 sin θ cos θ

⇒ 3 (sin θ - cos θ) = 2 sin 2θ

⇒ [3 (sin θ - cos θ)]\(^{2}\) = (2 sin 2θ)\(^{2}\), [Squaring both sides]

⇒ 9 (sin\(^{2}\) θ - 2 sin θ cos θ + cos\(^{2}\) θ) = 4 sin\(^{2}\) 2θ

⇒ 9 (sin\(^{2}\) θ + cos\(^{2}\) θ - 2 sin θ cos θ) = 4 sin\(^{2}\) 2θ

⇒ 9 (1 - 2 sin θ cos θ) = 4 sin\(^{2}\) 2θ

⇒ 4 sin\(^{2}\) 2θ + 9 sin 2θ - 9 = 0

⇒ (4 sin 2θ - 3)(sin 2θ + 3) = 0

⇒ 4 sin 2θ - 3 = 0 or sin 2θ + 3 = 0

⇒ sin 2θ = ¾ or sin 2θ = -3

but sin 2θ = -3 is not possible.

Therefore, sin 2θ = ¾ = sin ∝ (say)

⇒ 2θ =  nπ + (-1)\(^{n}\) ∝, where, n = 0, ± 1, ± 2, ± 3, ± 4 ..... and sin ∝ = ¾

⇒ θ = \(\frac{nπ}{2}\) + (-1)\(^{n}\) \(\frac{∝}{2}\), where, n = 0, ± 1, ± 2, ± 3, ± 4 ..... and sin ∝ = ¾

Therefore, the required solution θ = \(\frac{nπ}{2}\) + (-1)\(^{n}\) \(\frac{∝}{2}\), where, n = 0, ± 1, ± 2, ± 3, ± 4 ..... and sin ∝ = ¾


2. Find general solution of the equation cos 4θ = sin 3θ.

Solution: 

cos 4θ = sin 3θ                                  

⇒ cos 4θ = cos (\(\frac{π}{2}\) - 3θ)

Therefore, 4θ = 2nπ ± (\(\frac{π}{2}\) - 3θ) 

Therefore, either 4θ = 2nπ + \(\frac{π}{2}\) - 3θ Or, 4θ = 2nπ - \(\frac{π}{2}\) + 3x

⇒ 7θ = (4n + 1)\(\frac{π}{2}\) or, θ = (4n - 1)\(\frac{π}{2}\)

⇒ θ = (4n + 1)\(\frac{π}{14}\) or, θ = (4n - 1)\(\frac{π}{2}\)

Therefore the general solution of the equation cos 4θ = sin 3θ are θ = (4n + 1)\(\frac{π}{14}\)and θ = (4n - 1)\(\frac{π}{2}\) , where, n = 0, ±1, ±2, …………………..

 Trigonometric Equations




11 and 12 Grade Math

From Problems on Trigonometric Equation to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. 4th Grade Mental Math on Roman Numerals | Roman Numerals Quiz

    Feb 23, 24 03:55 PM

    In 4th grade mental math on numbers, students can practice different questions on write the Hindu-Arabic numerals, write the Roman Numerals, comparison of roman numerals, addition of roman numerals.

    Read More

  2. 4th Grade Mental Math on Numbers | Mental Math 4th Grade with Answers

    Feb 23, 24 02:24 PM

    4th Grade Mental Math on Numbers
    In 4th grade mental math on numbers, students can practice different questions on numbers in figures, number name, place value, face value, comparison of number and formation of greatest and smallest…

    Read More

  3. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 23, 24 01:28 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  4. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  5. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More