Textbook Question
Work each problem.
Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.18
Textbook Question
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
sin 35°
Verified step by step guidance1
Identify the trigonometric identity that relates sine and cosine: \( \sin(\theta) = \cos(90^\circ - \theta) \).
Apply the identity to the given expression: \( \sin(35^\circ) = \cos(90^\circ - 35^\circ) \).
Simplify the expression inside the cosine function: \( 90^\circ - 35^\circ = 55^\circ \).
Rewrite the expression using the identity: \( \sin(35^\circ) = \cos(55^\circ) \).
Match \( \cos(55^\circ) \) with the corresponding expression in Column II.
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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. These identities, such as the Pythagorean identity, angle sum and difference identities, and co-function identities, are essential for simplifying expressions and solving trigonometric equations. Understanding these identities allows students to manipulate and relate different trigonometric functions effectively.
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Sine Function
The sine function is one of the primary trigonometric functions, defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is periodic with a range of [-1, 1] and is commonly used in various applications, including wave motion and oscillations. Knowing the values of sine for common angles (like 30°, 45°, and 60°) helps in evaluating expressions involving sine for other angles.
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Angle Relationships
Angle relationships, such as complementary and supplementary angles, play a crucial role in trigonometry. For instance, the sine of an angle is equal to the cosine of its complement (sin(θ) = cos(90° - θ)). Recognizing these relationships allows students to find equivalent expressions and solve problems more efficiently by transforming angles into more manageable forms.
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