Textbook Question
Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.
178°
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
1:58 minutesProblem 37
Textbook Question
Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.
―345°
Verified step by step guidance1
First, understand that the angle given is \(-345^\circ\), which is a negative angle measured clockwise from the positive x-axis in standard position.
To find the equivalent positive angle, add \(360^\circ\) to \(-345^\circ\) because angles coterminal with \(\theta\) differ by \(360^\circ\). So, calculate \(-345^\circ + 360^\circ = 15^\circ\).
Determine the quadrant where the angle \(15^\circ\) lies. Since \(15^\circ\) is between \(0^\circ\) and \(90^\circ\), it lies in Quadrant I.
Recall the signs of trigonometric functions in Quadrant I: sine, cosine, and tangent are all positive in this quadrant.
Therefore, the signs of \(\sin(-345^\circ)\), \(\cos(-345^\circ)\), and \(\tan(-345^\circ)\) are all positive because the angle is coterminal with \(15^\circ\) in Quadrant I.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Position of an Angle
An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise for positive angles and clockwise for negative angles. Understanding this helps locate the angle's terminal side on the coordinate plane.
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Coterminal Angles
Coterminal angles share the same terminal side but differ by full rotations of 360°. To find a positive coterminal angle for a negative angle like -345°, add 360° until the angle lies between 0° and 360°. This simplifies determining the quadrant and the signs of trigonometric functions.
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Signs of Trigonometric Functions by Quadrant
The signs of sine, cosine, and tangent depend on the quadrant where the terminal side lies. In Quadrant I, all are positive; in Quadrant II, sine is positive; in Quadrant III, tangent is positive; and in Quadrant IV, cosine is positive. Identifying the quadrant is essential to determine the sign of each function.
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