Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 30

Chapter 2, Problem 30

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)

Verified step by step guidance

Step 1: Identify the coordinates of the point given on the terminal side of the angle \( \theta \). Here, the point is \( (-2\sqrt{3}, 2) \).

Step 2: Calculate the distance \( r \) from the origin to the point using the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-2\sqrt{3})^2 + 2^2} \]. This will be used to find the trigonometric functions.

Step 3: Determine the quadrant where the point lies. Since \( x = -2\sqrt{3} < 0 \) and \( y = 2 > 0 \), the point is in the second quadrant. This helps in determining the signs of the trigonometric functions.

Step 4: Find the six trigonometric functions using the definitions: \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}, \quad \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]. Substitute the values of \( x \), \( y \), and \( r \) into these formulas.

Step 5: Rationalize the denominators of the trigonometric function values where necessary to express the answers in simplest form.

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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 is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise by the angle measure θ. Understanding this helps in visualizing and sketching the angle based on a given point on its terminal side.

Finding the Reference Angle and Quadrant

The given point (―2√3, 2) lies in the second quadrant because x is negative and y is positive. The reference angle is the acute angle formed between the terminal side and the x-axis. Identifying the quadrant and reference angle is essential to determine the least positive measure of θ and the signs of trigonometric functions.

Six Trigonometric Functions from Coordinates

The six trigonometric functions (sin, cos, tan, csc, sec, cot) can be found using the coordinates of the point on the terminal side. First, calculate the radius r = √(x² + y²). Then, sin θ = y/r, cos θ = x/r, and tan θ = y/x. The reciprocal functions are csc θ = r/y, sec θ = r/x, and cot θ = x/y. Rationalizing denominators ensures simplified exact values.

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Related Practice

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. (―8 , 15)

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

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Textbook Question

Find the measure of each marked angle. See Example 2 supplementary angles with measures 6𝓍 - 4 and 8𝓍 - 12 degrees

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Textbook Question

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2.

17° 41' 13" , 96° 12' 10"

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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