Trigonometry

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6

SOLVING RIGHT TRIANGLES


This is a topic in traditional trigonometry. It does not come up in calculus.

To SOLVE A TRIANGLE means to know all three sides and all three angles.  When we know the ratios of the sides, we use the method of similar figures.  That is the method to use when solving an isosceles right triangle or a 30°-60°-90° triangle.  When we do not know the ratio numbers, then we must use the Table of ratios.  The following example illustrates how.

The general method

Example 1.   Given an acute angle and one side.   Solve the right triangle ABC if angle A is 36°, and side c is 10 cm.

Angle A = 36 degrees

Solution.  Since angle A is 36°, then angle B is 90° − 36° = 54°.

To find an unknown side, say a, proceed as follows:

1.   Make the unknown side the numerator of a fraction, and make the known side the denominator.
Unknown
 Known
 =   a 
10
2.   Name that function of the angle.
Unknown
 Known
 =   a 
10
 = sin 36°
3.   Use the trigonometric Table to evaluate that function.
Unknown
 Known
 =   a 
10
 = sin 36° = .588
4.   Solve for the unknown side.

a = 10 × .588 cm = 5.88 cm

(Lesson 4 of Arithmetic.)

Problem 1.   Solve the triangle for side b.

solve triangle

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To consult the Table, click here.

Unknown
 Known
 =   b 
10
 = cos 36° = .809
 
    b  = 10 × .809 = 8.09 cm

Problem 2.  To measure the width of a river.   Two trees stand opposite one another, at points A and B, on opposite banks of a river.

The width of a river

Distance AC along one bank is perpendicular to BA, and is measured to be 100 feet.  Angle ACB is measured to be 79°.  How far apart are the trees; that is, what is the width w of the river?

Unknown
Known  
 =   w 
100
 =  tan 79°.
  w  =  100 tan 79°
   =  100 × 5.145 = 514.5 ft,

from the Table.

(To measure the height of a flagpole, and for the meaning of the angle of elevation, see the Example in Topic 3.)

Example 2.   Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°.

solve triangle

Solution.   The angle of depression is the angle below straight ahead -- horizontal -- that an oberver must look in order to see something below the observer.  Thus in order to see the boat, the lighthouse keeper must look down 6°.

Now, the triangle formed by the lighthouse and the distance d of the boat from the lighthouse, is right-angled.  And since the angle of depression is 6°, then the alternate angle is also 6°. (Euclid, I. 29.)

If d is the distance of a boat from the lighthouse, then

  d  
100
= cot 6° = 9.514,  from the Table.

Therefore,

d = 951.4 meters.

Example 3.  Given two sides of a right triangle.   Solve the right triangle ABC given that side c = 25 cm and side b = 24 cm.

solve triangle

Solution.  To find the remaining side a, use the Pythagorean theorem:

a2 + 242  =  252
a2  =  625 − 576 = 49
a  =  square root of 49 = 7.

Next, to find angle A, we have

cos A =  24
25
  =    96
100
,  on multiplying each term by 4.
 
       =   .96  

(See Skill in Arithmetic:  Fractions into decimals.)

We must now inspect the Table to find the angle whose cosine is closest to .96, or, since this is a three place Table, .960.

We find

cos 16° = .961

Therefore,

Angle A approximately 16°.

Finally,

Angle B = 90° − 16° = 74°.

We have solved the triangle.

Problem 3.   Solve the right triangle ABC given that c = 10 cm and b = 8 cm.

solve triangle

To find the remaining side a, use the Pythagorean theorem:

a2 + 82  = 102
a2  =  100 − 64 = 36
a  =  square root of 36 = 6 cm.

To find angle A, we have

cos A =   8 
10
 = .8.

Now inspect the Table to find the angle whose cosine is closest to .8, or, since this is a three place Table, .800.

Find  cos 37° = .799.

Therefore, Angle Aapproximately37°.   Angle B = 90° − 37° = 53°.


Next Topic:  The Law of Cosines


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