Textbook Question
Use identities to correctly complete each sentence.
If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.44
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
Verified step by step guidance1
Recognize that \( \sec^2 x \) is the square of the secant function, which is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \).
Use the identity \( \sec^2 x = 1 + \tan^2 x \) to express \( \sec^2 x \) in terms of tangent.
Substitute \( \sec^2 x = 1 + \tan^2 x \) into the expression: \( 1 - \frac{1}{\sec^2 x} \).
Simplify the expression: \( 1 - \frac{1}{1 + \tan^2 x} \).
Recognize that the expression simplifies to a constant or a simpler trigonometric function by evaluating the fraction.
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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. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, serve as the foundation for simplifying trigonometric expressions. Understanding these identities is crucial for manipulating and simplifying expressions like the one given.
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Fundamental Trigonometric Identities
Reciprocal Functions
Reciprocal functions are pairs of trigonometric functions that are defined as the reciprocal of each other. For example, the secant function (sec x) is the reciprocal of the cosine function (cos x), meaning sec x = 1/cos x. Recognizing these relationships allows for easier simplification of expressions involving secant and cosine, as seen in the expression 1 - 1/sec² x.
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3:23Secant, Cosecant, & Cotangent on the Unit Circle
Simplification of Expressions
Simplification of expressions in trigonometry involves rewriting complex expressions in a more manageable form, often using identities. This process can include combining like terms, factoring, or substituting equivalent expressions. In the context of the given expression, simplifying 1 - 1/sec² x requires applying the reciprocal identity and recognizing that sec² x relates to cos² x, leading to a clearer expression.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
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Textbook Question
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
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Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot α sin α
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Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cot² θ(1 + tan² θ)
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