Multiple Choice
Find given , , & is in Q IV and is in Q II.
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.10
Textbook Question
Find the exact value of each expression.
sin 255°
Verified step by step guidance1
Step 1: Recognize that 255° is not a standard angle, so we need to express it in terms of known angles.
Step 2: Use the angle subtraction identity for sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
Step 3: Express 255° as 270° - 15°, which are angles we can work with.
Step 4: Apply the identity: \( \sin(255°) = \sin(270° - 15°) = \sin 270° \cos 15° - \cos 270° \sin 15° \).
Step 5: Substitute the known values: \( \sin 270° = -1 \), \( \cos 270° = 0 \), and use the values for \( \cos 15° \) and \( \sin 15° \) from trigonometric tables or identities.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle
The unit circle is a fundamental concept in trigonometry that defines the sine, cosine, and tangent functions based on the coordinates of points on a circle with a radius of one. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine and the y-coordinate represents the sine of that angle. Understanding the unit circle allows for the determination of exact values for trigonometric functions at various angles.
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Reference Angles
A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For angles greater than 180°, like 255°, the reference angle helps simplify the calculation of trigonometric values by relating them to angles in the first quadrant. In this case, the reference angle for 255° is 255° - 180° = 75°, which can be used to find the sine value.
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Sine Function Properties
The sine function is periodic and has specific properties based on the quadrant in which the angle lies. For angles in the third quadrant, such as 255°, the sine value is negative. Knowing that sin(θ) = -sin(reference angle) for angles in the third quadrant allows us to determine that sin(255°) = -sin(75°), leading to the exact value calculation.
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