PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
8th Edition
ISBN: 9781337904285
Author: Larson
Publisher: CENGAGE L
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Chapter 5.5, Problem 54E
To determine

Tofind:The exact values of the sine, cosine and tangent at angle π12 using half-angle formula.

Expert Solution & Answer
Check Mark

Answer to Problem 54E

The exact values of the sine, cosine and tangent at angle π12 are 232 , 2+32 and 23 , respectively.

Explanation of Solution

Given information:

The given angle is π12 .

Calculation:

Since, the angle π12 lies in Quadrant I. So, the values of sine, cosine and tangent will be positive.

The expression sinπ12 can be rewritten as,

  sin(π12)=sin(12π6)

Write the expression to calculate the value of sin(12π6) .

  sin(12π6)=±12(1cosπ6)=±12(1(32)) {cosπ6=32}

Since, the value of sine is positive in Quadrant I. So, consider the positive value and simplify it.

  sin(12π6)=232sinπ12=232

Write the expression to calculate the value of cosπ12 .

  cosπ12=cos(12π6)=±1+cosπ62=±1+(32)2 {cosπ6=32}

Since, the value of cosine is positive in Quadrant I. So, consider the positive value and simplify it.

  cosπ12=1+(32)2=2+32×2=2+32

Similarly, write the expression to calculate the value of tanπ12 .

  tanπ12=tan(12tanπ6)=±(1cosπ6sinπ6)=±(1(32)(12)) {cosπ6=32sinπ6=12}

Since, the value of tangent is positive in Quadrant I. So, consider the positive value and simplify it.

  tanπ12=1(32)(12)=(232)(12)=23

Therefore, the exact values of the sine, cosine and tangent at angle π12 are 232 , 2+32 and 23 , respectively.

Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Prob. 36ECh. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - Prob. 51ECh. 5.1 - Prob. 52ECh. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.1 - Prob. 59ECh. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.1 - Prob. 62ECh. 5.1 - Prob. 63ECh. 5.1 - 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How to apply the law of sines to find the remaining parts of a triangle; Author: Brian McLogan;https://www.youtube.com/watch?v=NdRF18HWkmE;License: Standard YouTube License, CC-BY