/*! 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 17 Find the exact values of the sin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 17

Find the exact values of the sine, cosine, and tangent of the angle. $$105^{\circ}=60^{\circ}+45^{\circ}$$

Short Answer

Expert verified
The exact values for sin, cos and tan of 105掳 are: \(\sin(105^{\circ})=\sqrt{6}+\sqrt{2}\), \(\cos(105^{\circ})=\sqrt{6}-\sqrt{2}\), \(\tan(105^{\circ})=2+\sqrt{3}\).

Step by step solution

01

Identifying the given angles

Establish that the angle of 105掳 is equal to the sum of angles 60掳 and 45掳.
02

Sine Rule for the sum of angles

Use the Sine addition rule, i.e. \(\sin(a+b) = \sin a \cdot \cos b + \cos a \cdot \sin b\), to find the sin of 105掳: \(\sin(105^{\circ})=\sin(60^{\circ}+45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}\).
03

Cosine Rule for the sum of angles

Next, apply the Cosine addition rule, i.e. \(\cos(a+b) = \cos a \cdot \cos b - \sin a \cdot \sin b\), to find the cos of 105掳: \(\cos(105^{\circ})=\cos(60^{\circ}+45^{\circ})=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}\).
04

Tangent Rule for the sum of angles

Lastly, employ the Tangent addition rule, i.e. \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b}\), to find the tan of 105掳: \(\tan(105^{\circ})=\tan(60^{\circ}+45^{\circ})=\frac{\tan60^{\circ}+\tan45^{\circ}}{1-\tan60^{\circ}\tan45^{\circ}}\).
05

Calculation

Now simply, calculate the sin, cos and tan values of 60掳 and 45掳: \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), \(\cos60^{\circ}=\frac{1}{2}\), \(\tan60^{\circ}=\sqrt{3}\), \(\sin45^{\circ}=\frac{1}{\sqrt{2}}\), \(\cos45^{\circ}=\frac{1}{\sqrt{2}}\) and \(\tan45^{\circ}=1\). Substitute these values back into the formulas obtained in step 2, 3 and 4 to get the exact values.
06

Final result

The exact values are: \(\sin(105^{\circ})=\sqrt{6}+\sqrt{2}\), \(\cos(105^{\circ})=\sqrt{6}-\sqrt{2}\) and \(\tan(105^{\circ})=2+\sqrt{3}\).

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

Fill in the blank. \(\cos (u+v)=\)_____

Fill in the blank. \(\cos (u-v)=\)_____

Find the \(x\) -intercepts of the graph. $$y=\sec ^{4}\left(\frac{\pi x}{8}\right)-4$$

Write the expression as the sine, cosine, or tangent of an angle. $$\cos 130^{\circ} \cos 40^{\circ}-\sin 130^{\circ} \sin 40^{\circ}$$

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in minutes) can be modeled by \(h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)\) The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.