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 79

Chapter 1, Problem 79

Use reference angles to find the exact value of each expression. Do not use a calculator. cot 19𝜋/6

Verified step by step guidance

First, recognize that the angle given is in radians: \(19\pi/6\). Since the trigonometric functions are periodic, reduce the angle to an equivalent angle between \(0\) and \(2\pi\) by subtracting multiples of \(2\pi\).

Calculate how many full rotations of \(2\pi\) fit into \(19\pi/6\). Since \(2\pi = 12\pi/6\), subtract \(12\pi/6\) from \(19\pi/6\) to get the reference angle within one full rotation: \(19\pi/6 - 12\pi/6 = 7\pi/6\).

Identify the quadrant where the angle \(7\pi/6\) lies. Since \(\pi = 6\pi/6\), \(7\pi/6\) is just past \(\pi\), so it lies in the third quadrant.

Find the reference angle for \(7\pi/6\) by subtracting \(\pi\): Reference angle \(= 7\pi/6 - \pi = 7\pi/6 - 6\pi/6 = \pi/6\).

Use the reference angle \(\pi/6\) to find \(\cot(\pi/6)\), then determine the sign of \(\cot(7\pi/6)\) based on the quadrant (third quadrant). Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and that both sine and cosine are negative in the third quadrant, so cotangent is positive there.

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.

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to an angle between 0° and 90° (or 0 and π/2 radians). Using reference angles allows you to find exact trigonometric values without a calculator.

Cotangent Function

Cotangent is the reciprocal of the tangent function, defined as cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). Understanding cotangent's relationship to sine and cosine is essential for evaluating its exact value, especially when using reference angles and known trigonometric values.

Angle Reduction and Coterminal Angles

Angles larger than 2π radians can be reduced by subtracting multiples of 2π to find a coterminal angle within one full rotation. This simplification helps identify the reference angle and the quadrant, which determines the sign of the trigonometric function.

Recommended video:

04:46

Coterminal Angles

Related Practice

Textbook Question

Find the area of the sector of a circle of radius r formed by a central angle θ. Express area in terms of π. Then round your answer to two decimal places. Radius, r: 4 inches Central Angle, θ: θ = 240°

Textbook Question

In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

views

Textbook Question

Use reference angles to find the exact value of each expression. Do not use a calculator. sec 495°

views

Textbook Question

Express each angular speed in radians per second. 6 revolutions per second

Textbook Question

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

Textbook Question

views