EBK PRECALCULUS W/LIMITS
EBK PRECALCULUS W/LIMITS
4th Edition
ISBN: 9781337516853
Author: Larson
Publisher: CENGAGE CO
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Chapter 5.5, Problem 36E
To determine

Tofind:The exact value of trigonometric ratios sin1650 , cos1650 , tan1650 using half - angle formulae.

Expert Solution & Answer
Check Mark

Answer to Problem 36E

Therequired values of trigonometric ratios are sin1650=0.2588 , cos1650=0.9659 , and tan1650=0.2680 .

Explanation of Solution

Given information:

The angle A=1650 .

Formulaused:

  cosA=2cos2A21 , cosA=12sin2A2 , and tanA=2tan(A/2)1tan2(A/2) .

Calculations:

We have cosA=2cos2A21cosA2=1+cosA2 ..................(1)

Now, substituting A=3300 in result (1) we have

  cos33002=1+cos33002cos1650=1+cos33002

Now, cos3300=cos(3600300)=cos(300)=32=0.8660 ,

  cos1650=1+cos33002=±1+0.86602=±0.9659

Since angle 1650 falls in second quadrant where cos1650 is negative, therefore, we accept cos1650=0.9659 .

We have cosA=12sin2A2sinA2=1cosA2 ..................(2)

Now, substituting A=3300 in result (2) we have

  sin3302=1cos3302sin1650=1cos33002 .

  sin1650=1(0.8660)2=10.86602=±0.2588

Since angle 1650 falls in second quadrant where sin1650 is positive, therefore, we accept sin1650=0.2588 .

We have tanA=2tan(A/2)1tan2(A/2)..................(3)

Now, substituting A=3300 in result (3) we have

  tan3300=2tan(3300/2)1tan2(3300/2)tan3300=2tan16501tan216500.57735=2tan16501tan21650

Suppose x=tan1650 , then 0.57735=2x1x2

  1x2=2x0.57735=3.4641xx23.4641x1=0 ..................(4)

Now, we solve quadratic equation (4) with formula x=b±b24ac2a for quadratic polynomial ax2+bx+c=0 as

  x=(3.4641)±(3.4641)24(1)(1)21=3.4641±12+42x=3.4641±42=3.7320,0.2680 ,

Since angle 1650 falls in second quadrant where tan1650 is positive, therefore, we accept x=0.2680 .Thus, tan1650=0.2680 .

Conclusion:

Thevalues of trigonometric ratios are sin1650=0.2588 , cos1650=0.9659 , and tan1650=0.2680 .

Chapter 5 Solutions

EBK PRECALCULUS W/LIMITS

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Prob. 34ECh. 5.5 - Prob. 35ECh. 5.5 - Prob. 36ECh. 5.5 - Prob. 37ECh. 5.5 - Prob. 38ECh. 5.5 - Prob. 39ECh. 5.5 - Prob. 40ECh. 5.5 - Prob. 41ECh. 5.5 - Prob. 42ECh. 5.5 - Prob. 43ECh. 5.5 - Prob. 44ECh. 5.5 - Prob. 45ECh. 5.5 - Prob. 46ECh. 5.5 - Prob. 47ECh. 5.5 - Prob. 48ECh. 5.5 - Prob. 49ECh. 5.5 - Prob. 50ECh. 5.5 - Prob. 51ECh. 5.5 - Prob. 52ECh. 5.5 - Prob. 53ECh. 5.5 - Prob. 54ECh. 5.5 - Prob. 55ECh. 5.5 - Prob. 56ECh. 5.5 - Prob. 57ECh. 5.5 - Prob. 58ECh. 5.5 - Prob. 59ECh. 5.5 - Prob. 60ECh. 5.5 - Prob. 61ECh. 5.5 - Prob. 62ECh. 5.5 - Prob. 63ECh. 5.5 - Prob. 64ECh. 5.5 - Prob. 65ECh. 5.5 - Prob. 66ECh. 5.5 - Prob. 67ECh. 5.5 - Prob. 68ECh. 5.5 - Prob. 69ECh. 5.5 - Prob. 70ECh. 5.5 - Prob. 71ECh. 5.5 - Prob. 72ECh. 5.5 - Prob. 73ECh. 5.5 - Prob. 74ECh. 5.5 - Prob. 75ECh. 5.5 - Prob. 76ECh. 5.5 - Prob. 77ECh. 5 - Prob. 1RECh. 5 - Prob. 2RECh. 5 - Prob. 3RECh. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - Prob. 6RECh. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - 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