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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 1 - Trigonometric FunctionsProblema 31
Capitolo 2, Problema 31

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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1
Identify the quadrant in which the angle 178° lies. Since 178° is between 90° and 180°, it lies in the second quadrant.
Recall the signs of the trigonometric functions in the second quadrant: sine (sin) is positive, cosine (cos) is negative, and tangent (tan) is negative.
Use the reference angle to understand the values better. The reference angle for 178° is calculated as \(180^\circ - 178^\circ = 2^\circ\).
Determine the signs of the reciprocal functions based on the signs of sine, cosine, and tangent: cosecant (csc) has the same sign as sine, secant (sec) has the same sign as cosine, and cotangent (cot) has the same sign as tangent.
Summarize the signs: sin(178°) > 0, cos(178°) < 0, tan(178°) < 0, csc(178°) > 0, sec(178°) < 0, cot(178°) < 0.

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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 rotates counterclockwise for positive angles. Understanding this helps locate the angle's terminal side on the coordinate plane, which is essential for determining the signs of trigonometric functions.
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Drawing Angles in Standard Position

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each determining the sign of sine, cosine, and tangent. For example, in the second quadrant (90° to 180°), sine is positive while cosine and tangent are negative. Knowing the quadrant of the angle helps identify the signs of its trigonometric values.
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Quadratic Formula

Reference Angle

The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps find the exact values of trigonometric functions by relating them to known angles in the first quadrant, while the quadrant determines the sign. For 178°, the reference angle is 2°.
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Reference Angles on the Unit Circle
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