Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 8

Chapter 1, Problem 8

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (-1, -3)

Verified step by step guidance

Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \((-1, -3)\), so \(x = -1\) and \(y = -3\).

Calculate the radius \(r\), which is the distance from the origin to the point, using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values to get \( r = \sqrt{(-1)^2 + (-3)^2} \).

Recall the definitions of the six trigonometric functions in terms of \(x\), \(y\), and \(r\): \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}, \quad \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]

Substitute the values of \(x\), \(y\), and \(r\) into each of the six functions to express them exactly in terms of radicals and integers.

Simplify each expression if possible, and pay attention to the signs of the functions based on the quadrant where the point \((-1, -3)\) lies (which is the third quadrant).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Coordinates and the Terminal Side of an Angle

The terminal side of an angle θ in standard position passes through a point (x, y). This point helps determine the angle's trigonometric values by relating x and y to the radius (r), which is the distance from the origin to the point.

Radius (r) and Its Calculation

The radius r is the distance from the origin to the point (x, y) on the terminal side, calculated using the Pythagorean theorem: r = √(x² + y²). It is essential for normalizing the coordinates when finding trigonometric function values.

Six Trigonometric Functions and Their Definitions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined using x, y, and r: sin θ = y/r, cos θ = x/r, tan θ = y/x, and their reciprocals csc θ, sec θ, cot θ. Knowing these definitions allows exact value computation from coordinates.

Recommended video:

6:04

Introduction to Trigonometric Functions

Related Practice

Textbook Question

In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 10 inches Arc Length, s: 40 inches

Textbook Question

In Exercises 8–12, draw each angle in standard position. 5𝜋 6

views

Textbook Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

<IMAGE>

cos 5𝜋/6

views

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

cos 2𝜋/3

Textbook Question

In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 6 yards Arc Length, s: 8 yards

views

Textbook Question

In Exercises 8–13, find the exact value of each expression. Do not use a calculator. tan 300°

views