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Chapter 5: Problem 103

Explain in your own words how knowledge of algebra is important when solving trigonometric equations.

Short Answer

Expert verified
Algebra is critical in solving trigonometric equations. It is used in simplifying, expanding and factorizing equations, solving for unknowns, and substituting trigonometric identities. Through these algebraic manipulations, trigonometric equations can be made simpler to solve.

Step by step solution

01

Understanding Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles. The three basic trigonometric functions that are typically introduced at high school are sine, cosine and tangent, often with the use of right angled triangles. Trigonometric equations often involve these functions along with their inverses and identities.
02

Understanding Algebra

Algebra involves the use of symbols (like x or y) to represent numbers. It allows us to create formulas and equations, and solve these equations by various methods like factorizing, using the quadratic formula or simplifying.
03

Algebraic Manipulation in Trigonometry

When you are dealing with a trigonometric equation, several algebraic manipulations can be used. You may need to expand expressions, combine like terms, factorize, or use the quadratic formula. For example, if you have an equation like \(\sin(x) + \sin(x) = 1\), you can combine like terms to simplify this to \(2\sin(x) = 1\). Then, you can solve for x using algebraic methods by dividing both sides by 2 to find \(\sin(x) = 0.5\). Then using inverse trigonometric functions (or tables or calculator), we can find the value of x.
04

Importance of Algebra in Trigonometry

Algebra is important in solving trigonometric equations because it enables us to manipulate the equation into a form that can be solved more easily. By simplifying, factorizing, expanding or rearranging, we can solve the trigonometric equation. It also helps in substitution of trigonometric identities and solving for unknowns. These are critical steps in trigonometric equation solving process and hence, knowledge of algebra is very important

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Most popular questions from this chapter

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \sin^4 x \cos^4 x \)

In Exercises 129 and 130, graph the function by hand in the interval \(\left[0,2\pi\right] \) by using the power-reducing formulas. \( f(x) = \cos^2 x \)

In Exercises 103 - 106, find all solutions of the equation in the interval \( \left[0,2\pi\right) \). Use a graphing utility to graph the equation and verify the solutions. \( \cos 2x - \cos 6x = 0 \)

In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \sin u = - \dfrac{3}{5}, \dfrac{3\pi}{2} < u < 2\pi \)

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin 5 \theta \sin 3 \theta \)
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