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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 1 - Trigonometric FunctionsProblem 31
Chapter 2, Problem 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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Key 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 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
Related Practice
Textbook Question

Find the measure of each marked angle. See Example 2 complementary angles with measures 3𝓍 ― 5 and 6𝓍 ― 40 degrees

Textbook Question

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (6√3 , ―6)

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Textbook Question

Find the measure of the smaller angle formed by the hands of a clock at the following times.

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Textbook Question

Find the measure of each marked angle. See Example 2 complementary angles with measures 9𝓍 + 6 and 3𝓍 degrees

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Textbook Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

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Textbook Question

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , y/r

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