Question

What values of $x$ satisfy the equation $\sin x=\cos x ?$

   What values of $x$ satisfy the equation $\sin x=\cos x ?$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 6, Problem 88

Instant Answer

Solved by Expert Joshua Eastwood

04/12/2020

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We can divide both sides by $\cos x$ to get $\tan x = 1$. Show more…

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What values of $x$ satisfy the equation $\sin x=\cos x ?$

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Key Concepts

Special Angles

Special angles are those for which the values of trigonometric functions are well-known and can be expressed exactly. Recognizing these angles simplifies the process of verifying solutions in trigonometric equations and aids in constructing general solution sets.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, describe the relationship between an angle and ratios of sides in a right triangle, and they are also used to model periodic phenomena. Their fundamental properties, including amplitude, period, and phase shift, are essential for setting up and solving equations involving these functions.

Trigonometric Equations

Trigonometric equations involve finding all angle values that satisfy a relationship between trigonometric functions. Solving these equations typically requires the application of identities, inverse trigonometric functions, and an understanding of the functions’ periodic behaviors.

Tangent Function

The tangent function, defined as the ratio of the sine function to the cosine function, becomes particularly useful when dealing with equations where sine and cosine are equated. Converting such equations into a tangent form often simplifies the problem by reducing it to solving for when the tangent takes a specific value.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. This property is crucial when solving trigonometric equations because it allows the derivation of a general solution that includes an infinite number of angle measures, each separated by a fixed period.

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