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Australia
Year 10

5.03 Degrees, minutes, and seconds

Lesson

Introduction

We've already looked at how to find angles and distances using trigonometric ratios, but have measured or rounded these angles to whole degrees.

Often angles are involved in measuring values where accuracy is important, such as location or distance. So we need to be more accurate and use smaller units to measure angles.

Degrees, minutes, and seconds

When talking about parts of an angle that are less than a degree, one option would be to use decimal values. For example, we can measure an angle to be 34.56\degree which means it is between 34\degree and 35\degree. We can also use different units of measurement to describe the smaller parts of an angle, and these units are minutes and seconds.

The image shows the conversion of degrees to minutes and minutes to seconds. Ask your teacher for more information.

There are 60 minutes in 1 degree. We write a minutes as a'.

There are 60 seconds in 1 minute. We write b seconds as b''.

The angle can then be written as ⬚\degree \, ⬚' \, ⬚''.

To convert a number in decimal form into minutes and seconds, we need first to find how much of 60 minutes the decimal part of the number represents.

For example, let's say we wanted to round 15.72 \degree to the nearest minute.

We need to work out 15\degree + 0.72 of a degree, which is 15 \degree plus 0.72 of 60 minutes.

0.72 \times 60=43.2 \text{ minutes}

Because we are rounding to the nearest minute, we then round the number of minutes to a whole number, which in this case is 43.

So, 15.72\degree rounded to the nearest minute is 15 \degree 43'.

But what if instead we wanted to round 15.72\degree to the nearest second?

Writing 15.72\degree in terms of degrees and minutes (without rounding) gave us 15 \degree 43.2'. To get the number of seconds we will need to look at the decimal part in the number of minutes and find out how many seconds out of 60 seconds it represents.

We need to work out 43' + 0.2 of a minute, which is 43 plus 0.2 of 60 seconds.

0.2 \times 60=12 \text{ seconds}

So, 15.72\degree rounded to the nearest minute is 15 \degree 43' 12''.

Examples

Example 1

Write 38.38\degree in degrees, minutes and seconds.

Worked Solution
Create a strategy

We multiply the decimal by 60 to convert it to minutes, and if any decimals remain in minutes, we do the same conversion for the seconds.

Apply the idea
\displaystyle 38.38\degree\displaystyle =\displaystyle 38\degree + 0.38 \times 60 \text{ minutes}Convert the decimal to minutes
\displaystyle =\displaystyle 38\degree 22.8'Evaluate the multiplication
\displaystyle =\displaystyle 38\degree 22' + 0.8 \times 60 \text{ seconds}Convert the decimal to seconds
\displaystyle =\displaystyle 38\degree 22'48''Evaluate the multiplication
Idea summary

There are 60 minutes in 1 degree. We write a minutes as a'.

There are 60 seconds in 1 minute. We write b seconds as b''.

To convert an angle in degrees to degrees, minutes, and seconds:

  • The whole number part is the number of degrees.

  • Multiply any decimal part by 60 to find the number of minutes.

  • Multiply any decimal part of the minutes by 60 to find the number of seconds.

  • The angle can then be written as ⬚\degree \, ⬚' \, ⬚''.

Note, you can also convert a decimal to degrees minutes and seconds using your calculator by pressing the button with \, \degree \, ' \, '' \, or \, \text{S}\leftrightarrow \text{D}\, on it.

Round degrees, minutes and seconds

Rounding values expressed in degrees and minutes is similar to rounding decimals. However, because there are 60 minutes in a degree, the half way point is 30:

  • If the number of minutes is less than 30, we round down to the nearest degree.

  • If the number of minutes is 30 or more, we round up to the nearest degree.

For example, when rounding to the nearest degree 148\degree 38'25'' becomes 149\degree and 148\degree27'25'' becomes 148\degree .

The same principle is true for rounding seconds to minutes - since there are 60 seconds in a minute, the half way point is 30:

  • If the number of seconds is less than 30, we round down to the nearest minute.

  • If the number of seconds is 30 or more, we round up to the nearest minute.

For example, when rounding to the nearest minute 71\degree13'45'' becomes 71\degree14' and 71\degree13'20'' becomes 71\degree 13'.

Examples

Example 2

Find the acute angle \theta. Round your answer to the nearest minute.

\sin \theta = 0.3168

Worked Solution
Create a strategy

Use the inverse sine function.

Apply the idea
\displaystyle \theta\displaystyle =\displaystyle \sin^{-1}(0.3168)Apply inverse sine both sides
\displaystyle \approx\displaystyle 18.47\degreeEvaluate using a calculator

Round the decimal to the nearest minute:

\displaystyle \theta\displaystyle =\displaystyle 18\degree + 0.47 \times 60 \text{ minutes}Convert the decimals to minutes
\displaystyle =\displaystyle 18\degree28.2'Evaluate the multiplication
\displaystyle \approx\displaystyle 18\degree28'Round the answer

Example 3

Use the side lengths provided to find the angle \theta to the nearest minute.

A right angled triangle A B C with angle theta at B, opposide side length of 60, and adjacent side length of 11.
Worked Solution
Create a strategy

Use the trigonometric ratio of tangent.

Apply the idea

With respect to the angle \theta, the opposite side is AC=60, and the adjacent side is BC=11, so we can use the tangent ratio.

\displaystyle \tan \theta\displaystyle =\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}Use the tangent ratio
\displaystyle \tan \theta\displaystyle =\displaystyle \frac{60}{11}Substitute the values
\displaystyle \theta\displaystyle =\displaystyle \tan^{-1}\left(\dfrac{60}{11}\right)Apply inverse tangent on both sides
\displaystyle \approx\displaystyle 79.611142 \degreeEvaluate using a calculator
\displaystyle =\displaystyle 79 \degree 36'\, 40.11''Convert using a calculator
\displaystyle =\displaystyle 79 \degree 37'Round to the nearest minute
Idea summary

To round to the nearest degree:

  • If the number of minutes is less than 30, we round down to the nearest degree.

  • If the number of minutes is 30 or more, we round up to the nearest degree.

To round to the nearest minute:

  • If the number of seconds is less than 30, we round down to the nearest minute.

  • If the number of seconds is 30 or more, we round up to the nearest minute.

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