/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Evaluate the sine, cosine, and t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 62

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$\frac{7 \pi}{6}$$

Short Answer

Expert verified
The sine of \(\frac{7 \pi}{6}\) is \(-\frac{1}{2}\), the cosine is \(-\frac{\sqrt{3}}{2}\), and the tangent is \(\frac{\sqrt{3}}{3}\).

Step by step solution

01

Identify the Reference Angle

The reference angle is obtained by subtracting multiples of \(\pi\) (or 180 degrees) from the given angle until the result is within the range of 0 to \(\pi/2\) (or 0 to 90 degrees). Here, subtract \(\pi\) from \(\frac{7 \pi}{6}\) to get \(\frac{1 \pi}{6}\), which is the reference angle.
02

Evaluate the Sine of the Angle

While the reference angle's sine is \(\frac{1}{2}\), the original angle is in the third quadrant where sine is negative. So the sine of \(\frac{7 \pi}{6}\) is \(-\frac{1}{2}\).
03

Evaluate the Cosine of the Angle

Similarly, the cosine of the reference angle is \(\frac{\sqrt{3}}{2}\). But since the original angle is in the third quadrant where cosine is negative, the cosine of the angle \(\frac{7 \pi}{6}\) is \(-\frac{\sqrt{3}}{2}\).
04

Evaluate the Tangent of the Angle

The tangent of an angle is the ratio of its sine to its cosine. Hence, the tangent of \(\frac{7 \pi}{6}\) is the sine of \(\frac{7 \pi}{6}\) divided by the cosine of \(\frac{7 \pi}{6}\), which is \(\frac{\frac{-1}{2}}{\frac{-\sqrt{3}}{2}}\). So, the tangent of \(\frac{7 \pi}{6}\) equals \(\frac{\sqrt{3}}{3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Finding the Central Angle Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an are of length \(s\). \(r=80\) kilometers, \(s=150\) kilometers

Determine whether the statement is true or false. Justify your answer. To find the reference angle for an angle \(\theta\) (given in degrees), find the integer \(n\) such that \(0 \leq 360^{\circ} n-\theta \leq 360^{\circ} .\) The difference \(360^{\circ} n-\theta\) is the reference angle.

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin \frac{x}{2}$$

Use a graphing utility to graph the function. $$f(x)=\arctan (2 x-3)$$

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{4} \sin 6 \pi t$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.