/*! 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 35 Trigonometric identities Find ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 35

Trigonometric identities Find the exact value of \(\cos (\pi / 12)\)

Short Answer

Expert verified
Answer: \(\cos(\pi / 12) = \frac{1}{4\sqrt{2}} (\sqrt{3} + 1)\)

Step by step solution

01

Express 蟺/12 as a sum of known angles

Express 蟺/12 as a sum of two angles which are multiples of 蟺/4 or 蟺/6. Let's express 蟺/12 as a sum of 蟺/4 and 蟺/6: \(\frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}\) Now we have A=蟺/4 and B= -蟺/6.
02

Apply the angle addition formula for cosine

Now we will use the angle addition formula for cosine with A = 蟺/4 and B = -蟺/6 to find the cosine of (蟺/12): \(\cos (\pi / 12) = \cos (\pi / 4 - \pi / 6) = \cos (\pi / 4) \cos (\pi / -6) - \sin (\pi / 4) \sin (\pi / -6)\)
03

Find the exact values of trigonometric functions for A and B

Compute the exact values of \(\cos (\pi / 4)\), \(\cos(\pi / -6)\), \(\sin (\pi / 4)\), and \(\sin (\pi / -6)\): \(\cos (\pi / 4) = \frac{1}{\sqrt{2}}\) \(\cos (-\pi / 6) = \cos (\pi / 6) = \frac{\sqrt{3}}{2}\) \(\sin (\pi / 4) = \frac{1}{\sqrt{2}}\) \(\sin (-\pi / 6) = -\frac{1}{2}\)
04

Plug the exact values into the angle addition formula for cosine

Substitute the computed values into the angle addition formula obtained in Step 2: \(\cos (\pi / 12) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{-1}{2}\)
05

Simplify the expression and find the value of cos(蟺/12)

Now we will simplify the expression: \(\cos (\pi / 12) = \frac{\sqrt{3}}{4\sqrt{2}} + \frac{1}{4\sqrt{2}}\) Next, factor out \(\frac{1}{4\sqrt{2}}\): \(\cos (\pi / 12) = \frac{1}{4\sqrt{2}} (\sqrt{3} + 1)\) Thus, the exact value of \(\cos (\pi / 12)\) is given by: \(\cos (\pi / 12) = \frac{1}{4\sqrt{2}} (\sqrt{3} + 1)\)

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

The velocity of a skydiver (in \(\mathrm{m} / \mathrm{s}\) ) \(t\) seconds after jumping from the plane is \(v(t)=600\left(1-e^{-k t / 60}\right) / k\), where \(k > 0\) is a constant. The terminal velocity of the skydiver is the value that \(v(t)\) approaches as \(t\) becomes large. Graph \(v\) with \(k=11\) and estimate the terminal velocity.

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabola \(y=x^{2}+2\) and the line \(y=x+4\)

Without using a calculator, evaluate or simplify the following expressions. $$\sec ^{-1} 2$$

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed. distance \(=\) speed \(\cdot\) time elapsed or \(d=v t\) A function \(y=f(x)\) such that if you run at a constant rate of \(5 \mathrm{mi} / \mathrm{hr}\) for \(x\) hours, then you run \(y\) miles

Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic \(y=f(x)=x^{3}+a x\). Find the inverse of the following cubics using the substitution (known as Vieta's substitution) \(x=z-a /(3 z) .\) Be sure to determine where the function is one-to-one. $$f(x)=x^{3}-2 x$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.