Textbook Question
Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using
a. difference identities
b. half-angle identities.
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.20
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
Recognize that the expression involves the sine function with a negative sign: \(-\sin 35^\circ\).
Recall the identity for the sine of a negative angle: \(\sin(-\theta) = -\sin(\theta)\).
Apply the identity to the given expression: \(-\sin 35^\circ = \sin(-35^\circ)\).
Understand that \(\sin(-35^\circ)\) is equivalent to the sine of a negative angle, which is a standard trigonometric identity.
Match \(\sin(-35^\circ)\) with the corresponding expression in Column II that represents the same value.
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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. They are essential for simplifying expressions and solving trigonometric equations. Common identities include the Pythagorean identities, angle sum and difference identities, and co-function identities, which relate the values of trigonometric functions at complementary angles.
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Co-function Identity
Co-function identities state that the sine of an angle is equal to the cosine of its complement. Specifically, sin(θ) = cos(90° - θ). This identity is useful for transforming expressions involving sine into cosine and vice versa, allowing for easier comparison and simplification of trigonometric expressions.
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Angle Measurement
Angle measurement in trigonometry can be expressed in degrees or radians. Understanding how to convert between these two systems is crucial for solving problems involving trigonometric functions. For example, 35° can be converted to radians, which may be necessary when using certain identities or when working with calculators that are set to radian mode.
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