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 4

Chapter 1, Problem 4

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. (3, 7)

Verified step by step guidance

Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (3, 7) \), so \( x = 3 \) and \( y = 7 \).

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 \( r = \sqrt{3^2 + 7^2} \).

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, but do not approximate the values numerically to maintain exactness.

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). These coordinates represent the position on the Cartesian plane, which helps determine the values of trigonometric functions based on the angle's location.

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios involving the coordinates (x, y) and the radius r = √(x² + y²). For example, sin(θ) = y/r and cos(θ) = x/r, linking geometry to trigonometry.

Calculating the Radius (r) from Coordinates

The radius r is the distance from the origin to the point (x, y), calculated using the Pythagorean theorem: r = √(x² + y²). This value is essential for finding the exact values of trigonometric functions since it normalizes the coordinates.

Recommended video:

02:13

Intro to Polar Coordinates Example 1

Related Practice

Textbook Question

Find a positive angle less than 2𝜋 that is coterminal with 16𝜋 3

views

Textbook Question

Find the reference angle for 16𝜋 3

views

Textbook Question

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 87.177°

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>

sin 𝜋/6

Textbook Question

In Exercises 2–4, convert each angle in degrees to radians. Express your answer as a multiple of 𝜋. 315°

Textbook Question

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

views