Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

2.2 Right-angled triangles – two known values, and one unknown value

Take a look at the following Worked example and Activity.

Worked example 2

Timing: Allow about 15 minutes

Figure 8 shows a right-angled triangle where the length of two sides are known, and just one angle – the right angle.

A right-angled triangle with one unknown side and two unknown angles.
Figure 8 A right-angled triangle with one unknown side and two unknown angles.

The aim is to calculate the values of α, β and b.

This time the length of two sides are known. In relation to the angle β, they are the adjacent and the hypotenuse.

the hypotenuse.The trig function which relates these two sides is cosine.

cosine of beta equals adjacent divided by hypotenuse
cosine of equation sequence beta equals 3.0 cm divided by 5.0 cm equals 0.6

This means that β is the angle for which cos is 0.6; this can be written as β = cos–1 (0.6).

Cos–1 is the opposite, or inverse, of taking the cosine of an angle and will give the angle (between 0 and 180°) with the cosine value of 0.6. Fortunately, your calculator also does this for you.

Look at your calculator again. You should see a key which has the word shift or the letters inv above it. This key enables you to use the inverse functions on a calculator. Look at your sin, cos and tan keys again. Above each you should see what are known as the inverse functions, sin–1, cos–1 and tan–1.

Again depending on the calculator, you have you will either need to key in the function first, or the number. Either way cos–1 (0.6) = 53.13010236. So, β = 53° (to 2 significant figures)

α can now be found as before:

α + β = 90°

α = 90° – β

α = 90° – 53.13010236°

α = 36° (to 2 significant figures)

The opposite side b can be found using Pythagoras.

multiline equation row 1 c squared equals a squared plus b squared row 2 five squared equals three squared plus b squared row 3 25 equals nine plus b squared row 4 25 minus nine equals b squared row 5 b squared equals 16 row 6 b equals 4.0 cm

So, the length of side b is 4.0 cm.

Now have a go yourself in this next activity.

Activity 4 Find the angles

Timing: Allow about 7 minutes

In the Figure 9, find the angles β and α and the length of side a.

A right-angled triangle for Activity 4.
Figure 9 A right-angled triangle for Activity 4

Discussion

The known sides are the opposite and hypotenuse. The trig function that links these is sine.

sine of beta equals opposite divided by hypotenuse
multiline equation row 1 sine of beta equals 5.0 postfix times cm divided by 8.0 postfix times cm row 2 beta equals sine super negative one of 5.0 postfix times cm divided by 8.0 postfix times cm row 3 beta equals 38.68 degree left parenthesis to two decimal places right parenthesis

and, since all the angles inside a triangle must add to 180°

α = 180° – (90° + 38.68°)

α = 51.32° (to 2 decimal places)

Now rounding to two significant figures to match the data in the question.

α = 51°

β = 39°

From Pythagoras:

multiline equation row 1 c squared equation left hand side equals right hand side a squared plus b squared row 2 a squared plus five squared equation left hand side equals right hand side eight squared row 3 a squared plus 25 equation left hand side equals right hand side 64 row 4 a squared equation left hand side equals right hand side 64 minus 25 row 5 a squared equation left hand side equals right hand side 39 row 6 a equation left hand side equals right hand side Square root of 39 row 7 equation left hand side equals right hand side 6.24 cm

The length of side a is 6.2 cm (to 2 significant figures).

So you know all the sides and all the angles in the triangle now.

Practicing new ideas and skills you have learned in maths is important, so here is one more activity for you to complete.

Activity 5 Finding sides and angle

Timing: Allow about 7 minutes

Look at Figure 10 and find the sides a and b and the angle α.

A right-angled triangle for Activity 5.
Figure 10 A right-angled triangle for Activity 5.

Discussion

The known sides are the opposite and hypotenuse. The trig function that links these is sine., but you still need to look for two known values related to one of the unknown values.

If you start by finding a (the adjacent), you need to use cosine as the adjacent and hypotenuse are related by that function.

cosine of beta equals adjacent divided by hypotenuse
multiline equation row 1 cosine of 45 degree equals a divided by 6.0 cm row 2 6.0 cm prefix multiplication of cosine of 45 degree equals a row 3 6.0 cm prefix multiplication of 0.7071 equals a row 4 side a equals 4.2 left parenthesis to two significant figures right parenthesis

You can now either use Pythagoras to work out b or the sine function.

multiline equation row 1 sine of 45 degree equals b divided by 6.0 cm row 2 6.0 cm prefix multiplication of sine of 45 degree equals b row 3 6.0 cm prefix multiplication of 0.7071 equals a row 4 side b equals 4.2 cm left parenthesis to two significant figures right parenthesis

You can check this using Pythagoras.

multiline equation row 1 c squared equals a squared plus b squared row 2 b squared equals c squared minus a squared row 3 b squared equals 6.0 squared minus 4.242640687 squared row 4 b squared equals 18 row 5 b equals Square root of 18 row 6 b equals 4.2 cm

This gives the same value as using sine as expected.

To find the other angle:

multiline equation row 1 alpha equals 180 degree minus open 90 degree plus 45 degree close row 2 alpha equals 180 degree minus 135 degree row 3 angle alpha equals 45 degree

This is a particular type of triangle where side a = side b and there are two equal angles.

This special sort of triangle is known as an isosceles triangle but is still also a right-angled triangle.

Trigonometry is powerful mathematical tool and the next section, before the weekly quiz, provides an example of trigonometry in practice.

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