8.06 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.

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

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\degree	Evaluate 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

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Example 3

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

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 \degree	Evaluate using a calculator
	\displaystyle =	\displaystyle 79 \degree 36'\, 40.11''	Convert using a calculator
	\displaystyle =	\displaystyle 79 \degree 37'	Round to the nearest minute

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