Amy Y.

asked • 10/16/14

Acute angle trigonometry

1) in triangle QRS, r=4.1cm, s=2.7cm, and R=88 degrees. Determine the measure of angle Q

2) A radio tower is supported by 2 wires on opposite sides. On the ground, the ends of the wire are 46.5 m apart. One wire makes a 62 degree angle with the ground, the other makes a 68 degree angle with the ground. Determine the height of the tower.

3) Stella decided to ski to a friends cabiN. She skiied 8.0km in the direction of N40 degrees E. She rested than skied S30 degrees E And arrived at the cabin. The cabin is 9.5km from Her home, as the crow flies. Determine the distance she travelled on the second leg of her trip.

Yohan C.

Hey Amy,

For first one, use Law of Sines:
 
Sin Q = Sin R = Sin S
    q         r           s

Sin R = Sin 88
r = 4.1

Sin S = ?
s = 2.7
[Sin (88)] * 2.7 = 4.1 * Sin S

[Sin (88)] * 2.7 = Sin S
     4.1

0.658135422671118 = Sin S

Take Sin -1 to get the angle S and then subtract angle R & S from 180 to get the measure of angle Q.
 
 
 
 
For second problem, draw the picture.  Base is 46.5 m, left wire with angle 62 & right wire with 68.
 
Draw the vertex (perpendicular line: height) from the base.  You will see the division of two triangles.  Triangle on the left side (62, 90, 28) and (68, 90, 22) for the right side. 
 
Let's call the height y.
 
Use Law of Sines again:
 
Sin 68    =    Sin 50
left wire         46.5
 
And it comes out to be 56.2814m.  From here, revisit right triangles and trig functions.
sine = opp / hyp     cos = adj / hyp     tan = opp / adj
Since we figured it out the length of left wire (hypotenuse), we can use sine function.
 
sine = opp / hyp   --->   sine = y (height) / length of left wire
sin 62 = y / 56.2814
0.8829475928589 = y / 56.2814   
 
 
 
 
For third problem, you have to draw picture again.  You will draw a line (8.0 km) that is 40 degrees above the horizontal (to the right) since North 40 degrees East.  After that, draw a perpendicular line to the ground (broken or dotted line).  From here, draw a line down toward the right (30 degrees) for some.  From this point, you draw a line that connects Her Home and Cabin (9.5). 
 
Let's call the first leg is a, the second leg is b, and the third leg is c.
 
Now you have the triangle to work with: one is very first one (40, 90, 50) (first leg of the trip), second one (30, 90, 60) (coming down from first leg).  But second leg is going little bit more farther down.  And finally, the distance between Her Home and the Cabin (third leg) 9.5.
 
From here, use Law of Sines to figure out the angle (where 3rd leg & 2nd leg meet) down below.
First angle (where 1st & 2nd leg meet) = 80 degrees  (90 - 40 + 30).
Sin 80 = Sin  C
   9.5        8
 
After you get the angle C, subtract 80 and this value from 180 to get angle A (where 1st & 3rd leg meet).  From that angle, use Law of Cosine:
 
Since you will get a & c, and all values for each of the angles, you will use this equation:
b = a2 + c2 - 2ac cos B   --->  b = second leg
 
I know it takes a lot of steps but I hope you get it.
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10/16/14

Kevin C.

tutor
Isn't N40°E, 40° clockwise from due North, the vertical?  Also, isn't the angle between the first and second route opposite the distance from start?  If I'm correct, then the Law of Sines is appropriate, not the Law of Cosines.
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10/16/14

Yohan C.

Hey Kev,
 
I didn't see word "clockwise" in the problem.  But from your statement, are you saying " N40°E, 40°" is from vertical line and not from horizontal line? if that is the case, are you saying that it will be 50 degrees above horizontal to start?
 
For second part or second leg, isn't she keep going eastward (She rested than skied S30 degrees E And arrived at the cabin)? and then go westward to her home? what do you mean by "opposite the distance from start?"
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10/16/14

Kevin C.

tutor
If the starting point is A, the angle is 40° clockwise from North to arrive at B.  She then goes 30° from the South, toward East.  Since the lines and parallel and the alternate interior angles are congruent, the angle at point B is 70°.  She travels along this path to the cabin.  So far, the triangle ABC has AB = 8.0 km., BC is unknown, and CA = 9.5.  Therefore, we can use the Law of Sines to find the length BC.   Remember, N40°E means 40° East of North.  (In other words, you are facing North (straight up) and going 40° toward  the East.  Using the same idea, S30°E, means she was facing South (straight down) and turned 30° toward the East.  That makes the angle there 70°).  I hope that helps to clear up the problem.
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10/16/14

Kevin C.

tutor
By the way, I like your approach to problem #2.  So much simpler and more elegant than mine.
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10/16/14

Kevin C.

tutor
I reread your comment.  The line CA is not due West and, therefore, the side CA is not horizontal.  Easy assumption to make. 
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10/16/14

Yohan C.

Of course it is not horizontal.  Picture tells all. 
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10/17/14

Kevin C.

tutor
Then there is not o 40-50-90 triangle in the picture,as you stated.
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10/17/14

Yohan C.

Of course, every angle is acute in this problem.
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10/17/14

1 Expert Answer

By:

Kevin C. answered • 10/16/14

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1.  Use the Sine Law:  (sinS)/s =(sinR)/r to find angle S: 
 
sinS = 2.7sin88/4.1.  So S ≈ 41°.  Since the sum of the angles is 180°, Q ≈ 51°.
 
2.  Draw the diagram in which the base is 46.5 from point A to point B.
     Let ∠A = 62° and ∠B = 68°.
     Draw the perpendicular from point C (the intersections of the two lines from points A and B).
     Let the horizontal side adjacent to point A = x and the horizontal side adjacent to point B = 46.5 - x.
 
Since we have two right triangles and we want to solve for h, the vertical segment, use:
 
tan 62°  = h/x and tan 68° = h/(46.5 - x).
 
Solve each for x:  x = h/tan62 and  - x = h/tan68 - 46.5, so x = 46.5 - h/tan68
 
Set the 2 equations equal:  46.5 - h/tan68 = h/tan62.  Solve for h:  h/tan68+h/tan62 = 46.5.
 
h(1/tan68 + 1/tan68) = 46.5.  So:  h = 46.5/(1/tan68+1/tan62) = 49.69m.
 
3.  Let A represent the starting point, B the turning point, and C the final point.
 
Using the directions given, we have ΔABC with AB = 8, AC = 9.5, and ∠B = 70° (40°+ 30°)
 
Using the Law of Sines, gives us the missing angles.  ∠A = 57.69° and ∠C = 52.31°.
 
Using the Law of Sines again, finds x ≈ 8.5 km.
 
I hope this  helps.
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