Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 1

Chapter 3, Problem 1

In Exercises 1–60, verify each identity.sin x sec x = tan x

Verified step by step guidance

Start by recalling the definitions of the trigonometric functions involved: \( \sin x \), \( \sec x \), and \( \tan x \).

Express \( \sec x \) in terms of \( \cos x \): \( \sec x = \frac{1}{\cos x} \).

Substitute \( \sec x \) in the left-hand side of the identity: \( \sin x \sec x = \sin x \cdot \frac{1}{\cos x} \).

Simplify the expression: \( \sin x \cdot \frac{1}{\cos x} = \frac{\sin x}{\cos x} \).

Recognize that \( \frac{\sin x}{\cos x} \) is the definition of \( \tan x \), thus verifying the identity: \( \sin x \sec x = \tan x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.

Reciprocal Functions

Reciprocal functions in trigonometry refer to pairs of functions where one function is the reciprocal of another. For example, the secant function (sec x) is the reciprocal of the cosine function (cos x), and the cosecant function (csc x) is the reciprocal of the sine function (sin x). Recognizing these relationships helps in transforming and simplifying trigonometric expressions.

Quotient Identity

The quotient identity in trigonometry states that the tangent function (tan x) is the ratio of the sine function (sin x) to the cosine function (cos x). This identity can be expressed as tan x = sin x / cos x. Understanding this identity is essential for verifying equations that involve tangent, as it allows for the substitution of sine and cosine functions.

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