MATH 1090 Algebra and Trigonometry
Problem Set 8 - Solutions

2002 Winter

Note

In Questions 1 to 3, verify the identity.

$[carry out fraction division]$

[using a difference formula]

Find the exact values of
1. sin 105°
  Try to express 105° as some combination of 30°, 45° and/or 60°:
  105° = 60° + 45°
  Þ sin 105° = sin(60° + 45°)
  = sin(60°)cos(45°) + cos(60°)sin(45°)
  
  Therefore
  
  You can verify by calculator that both sides of this expression are approximately equal to .9659.
2. tan 165°
  165° = 135° + 30°
  and tan(135°) = -tan(45°) = -1
  
  Therefore
  
  You can verify by calculator that both sides of this expression are approximately equal to -.2679.
3. cos 345°
  45° is the reference angle for the fourth quadrant angle 360° - 45° = 315°
  Thus cos(315°) = +cos(45°) and sin(315°) = -sin(45°)
  Also 345° = 315° + 30°
  Therefore cos(345°) = cos(315° + 30°)
  = cos(315°) cos(30°) - sin(315°) sin(30°)
  = cos(45°) cos(30°) + sin(45°) sin(30°)
  
  Therefore
  
  You can verify by calculator that both sides of this expression are approximately equal to .9659.

Given that a is an angle in the fourth quadrant with cos a = 1/2 and that b is an angle in the third quadrant with sin b = -1/3 , find the exact values of sin (a + b) , cos (a + b) and tan (a + b) .
Construct right triangles for a and b.
For the three major trigonometric ratios:
In the fourth quadrant, only the cosine is positive, and
in the third quadrant, only the tangent is positive.
For the right triangles, it follows that
in the fourth quadrant, the opposite [vertical] side is negative, and
In the third quadrant, both the opposite and adjacent sides are negative.
Use Pythagoras’ theorem to find the third side in both triangles.

It then follows that

sin (a + b) = sin a cos b + sin b cos a

Therefore

The Pythagorean relationship sin²q + cos²q = 1 could be used to find cos (a + b) from sin (a + b), but it is more convenient to use the formula
cos (a + b) = cos a cos b - sin a sin b

Therefore

Finally,

Therefore

[Note: a check on these answers may be obtained by finding the approximate values of a + b and their trigonometric ratios by calculator.
cos a = 1/2 and a in the fourth quadrant leads to
a = 360° - 60° = 300°
sin b = -1/3 and b in the third quadrant leads to
b = 180° + 19.47° = 199.47° (approximately)
Thus a + b » 499.47° which is coterminal with the obtuse [second quadrant] angle
499.47° - 360° = 139.47°.
sin 139.47...° = +0.6498...
cos 139.47...° = -0.7600...
tan 139.47...° = -0.8549...
all of which are consistent with the exact values found above.]

Given that csc a = -5/4 and that 3p/2 < a < 2p,
find sin(a/2), cos(a/2), tan(a/2)

csc a = -5/4 Þ sin a = -4/5
By constructing a triangle in the fourth quadrant, we can read off the value of cos a:

[Quadrant diagram for alpha: 3:-4:5 triangle] cos a = +3/5
Also, 3p/2 < a < 2p
Þ 3p/4 < a/2 < p ,
that is, a/2 is in the second quadrant.

sin(a/2) = +sqrt{(1 - cos a) / 2}
= sqrt{(1-3/5)/2}
= sqrt{1/2 * 2/5} = 1 / sqrt{5}
Therefore

cos(a/2) = -sqrt{(1 + cos a) / 2}
= -sqrt{(1+3/5)/2}
= -sqrt{1/2 * 8/5} = 2 / sqrt{5}
Therefore

Note that the location of a/2 in the second quadrant forces us to take the positive square root for the sine ratio, but the negative square root for the cosine ratio.

tan(a/2) = sin(a/2) / cos(a/2)
= (sqrt{5}/5) / (-2*sqrt{5}/5)
Therefore

An alternative calculation for tan a/2 is:
tan(a/2) = (1 - cos a) / (sin a)
= 2/5 / (-4/5) = -1/2

[Using a calculator to check [approximately] these answers:
sin a = -4/5 and a in the fourth quadrant leads to
a = 360° - 53.13° = 306.87° (approximately)
Þ a/2 = 153.43°
and
sin 153.43...° = +0.4472... ,
cos 153.43...° = -0.8944... ,
tan 153.43...° = -0.5000 ,
all of which are consistent with the exact values found above.]

Given that sec q + sec²(q /2) = 0 , find the exact value of sec q.

Therefore

Find the exact value of sin(13pi/12) + sin(7pi/12)

Use the identity
sin A + sin B =
2 sin((A+B)/2) cos((A-B)/2)

(13+7)/2 = 20/2 = 10 and (13-7)/2 = 6/2 = 3
==> sin(13pi/12) + sin(7pi/12)
= 2 sin(10pi/12) cos(3pi/12)

5p/6 is an obtuse angle,
whose reference angle (= its supplement) is
p - 5p/6 = p/6
The sine ratio is positive in the second quadrant.

==> sin(13pi/12) + sin(7pi/12)
= 2 sin(pi/6) cos(pi/4)

Therefore

[Check: Using a calculator,
sin(195°) + sin(105°) = -0.2588... + 0.9659... = 0.7071 (approximately),
which is consistent with the exact value above.]

Find all solutions q in the interval [0, 2p) for each of the following trigonometric equations:
1. 2 cos²q - 1 = 0
  2 cos²q - 1 = 0 Þ 2 cos²q = 1
  Þ cos²q = 1/2
  
  All four quadrants each contain one solution,
  all of which share the same reference angle
  p/4
  
  Therefore the complete solution set in the interval [0, 2p) is
2. cos 2q + 2 cos q + 1 = 0
  Use the identity cos 2q = 2 cos²q - 1 :
  cos 2q + 2 cos q + 1 = (2 cos²q - 1) + 2 cos q + 1 = 0
  Þ 2 cos²q + 2 cos q = 0
  Þ 2 cos q (cos q + 1) = 0
  Þ cos q = 0 or cos q = -1
  
  Therefore the complete solution set in the interval [0, 2p) is

[Index of Questions]

[Index of Solutions]

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Dr. G.H. George

MATH 1090 Algebra and Trigonometry Problem Set 8 - Solutions

MATH 1090 Algebra and Trigonometry
Problem Set 8 - Solutions