Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

2.1 Right angled triangles – one known side, and two known angles

Take a look at the following Worked example and Activity.

Worked example 1

Timing: Allow about 10 minutes

In the example in Figure 6 there are three unknown values, α, a and c, and three known values.

A right angle triangle for Example 1.
Figure 6 A right angle triangle for Example 1

The angle β is 30° and the side b is 6.0 cm. The right-angle is 90°.

The unknown angle α can be found using the fact that the sum of the internal angles of a triangle is 180°.

multiline equation row 1 alpha equals 180 degree minus open 90 degree plus 30 degree close row 2 equals 180 degree minus 120 degree row 3 equals 60 degree

Now calculate a, which is ‘adjacent’ to the angle β, which is 30°. The opposite side is 6.0 cm.

If you look at SOHCAHTOA, you will find that the function relating adjacent and opposite is tan.

tangent of beta equals opposite open b close divided by adjacent open a close
multiline equation row 1 a equals b postfix times tangent of 30 degree equals 6.0 postfix times cm divided by a row 2 a equals b postfix times tangent of 30 degree equals 6.0 postfix times cm row 3 a equals 6.0 postfix times prefix times times of cm divided by tangent of times 30 postfix degree row 4 equals 10.39 cm

The side a is 10.39 cm.

The value for a has shown to 4 significant figures, rather than rounded to 2 significant figures at this point, to avoid introducing rounding errors in the final answer.

The only remaining unknown is c and this can be found using Pythagoras Theorem.

multiline equation row 1 c squared equals a squared plus b squared row 2 c squared equals multiline equation line 1 open 10.39230485 close squared plus six squared row 3 c squared equals 108 plus 36 row 4 c squared equals 144 row 5 c equals Square root of multiline equation line 1 144 row 6 c equals 12 cm

The side c is 12 cm. Using the rules for a triangle, some basic trigonometry and Pythagoras, and knowing the length of one side and the size of one angle, the remaining angles and sides of the triangle have been found.

Now try this one yourself.

Activity 3 Find the unknown side

Timing: Allow about 7 minutes

Find the unknown side b and c and the angle α for Figure 7.

A right angle triangle for Activity 3.
Figure 7 A right angle triangle for Activity 3

Answer

You can start by finding angle α.

Since the right angle is 90°:

α + β = 90°

α = 90° – β

α = 90° – 40°

α = 50°

Now you can either find b or c.

Starting with c this time, which is the hypotenuse.

The function that links the adjacent (the known value) with the hypotenuse is cosine.

cosine of beta equals adjacent divided by hypotenuse
cosine of equation left hand side 40 postfix degree equals right hand side 3.0 postfix times cm divided by c

Rerranging to make c the subject:

c × cos 40° = 3.0 cm

So, c equals 3.0 cm divided by cosine of 40 degree

c = 3.916 cm

The opposite side b can be found using Pythagoras.

multiline equation row 1 c squared equals a squared plus b squared row 2 3.916 squared equals three squared plus b squared row 3 15.3367972 equals nine plus b squared row 4 15.3367972 minus nine equals b squared row 5 b squared equals 6.336793719 row 6 b equals 2.517 cm

The length of side b is 2.5 cm (to 2 significant figures) and c 3.9 cm (to 2 significant figures).

Rounding both to two significant figures matches the precision with which the known side was shown.

The next section will consider the situation where angles α and β are unknown, but the length of two sides are known.

MST_1

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