Evaluate sine, cosine and tangent expressions

Trigonometry: A definition

Trigonometry is the study of angles and the edges that they are made up from. It forms a very important part of a number of professional fields including science, engineering and other areas of applied math. However, trigonometry is also used often by builders, carpenters, surveyors, architects, town planners and many other professions.

So what's the big deal? 

Trigonometry allows us to work with angles and lines that form triangles, helping us to figure out properties of certain objects. Using trigonometry we can find how tall something is or how far away it is or what its angle of elevation is. On top of this, we can extend these ideas to countless other areas such as analysing projectiles and their velocities, wave motion, complex numbers and into 3 dimensions (like in the image above, where the angles of the robotic arm and their positions are found using trigonometry).

As an old area of math, dating back to early centuries, trigonometry has proven to be a fundamental element in our understanding of both the Earth and Space we live in.

The interactive below shows how the trigonometric ratios change with different angles.

Take a look at what happens to the tangent function when the angle is 90 degrees (you can change the scale to zoom in and out). What does this really mean?  What are the opposite and adjacent lengths? Similarly look at the cosine of 0 or the sine of 90.

If you want to see how this interactive works you can watch this short video.  Watch video

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Worked Examples

QUESTION 1

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QUESTION 4

Degrees minutes and seconds in angles

Angles as we know are measured in degrees.  But did you know that if we break an angle up into its smaller parts we get minutes and seconds.  

1 degree is made up of 60 minutes. And every 1 minute is made up of 60 seconds.

This means that this is a base $60$60 system.  Instead of a base $10$10 system for our decimal numbers, to move between the 'columns' multiply or divide by $60$60.  And to move 2 columns we move $3600$3600 ($60\times60$60×60).  

 

Conversions between notations

A base 60 system means that we need to be extra careful when converting between decimal version of angle and the DMS (degree, minute, second) notation.  

DMS to decimal

To convert from DMS notation to decimal we convert each part separately.

Decimal to DMS

• Firstly remove the whole number, this is the whole degrees.
• Then take the decimal and multiply by $60$60.  
• Remove that whole number, this is the number of minutes.
• Take the decimal component of that number and multiply by $60$60. This is the number of seconds.  Round this to the required decimal places as necessary.  

 

Examples

Question 1

Convert $35.5$35.5 degrees into degrees, minutes and seconds.  

Think: A common misconception is to think that the decimal point means that $35.5$35.5 degrees is the same as $35^\circ5'$35°5′ .  Its not!  

I do know that $0.5$0.5 of anything means half.  This is half of a degree which is $0^\circ30'$0°30′ , because there are $60'$60′ in a degree.

Do: $35.5^\circ$35.5°  is $35$35 whole degrees and $0.5$0.5 of a degree.  So, $35.5^\circ$35.5° is $35^\circ30'$35°30′.

 

Question 2

Convert $28.42$28.42° into DMS notation.

$28^\circ$28°			Firstly remove the whole number, this is the whole degrees.
$0.42\times60$0.42×60	$=$=	$25.2'$25.2′	Then take the decimal and multiply by $60$60
$0^\circ25'$0°25′			Remove that whole number, this is the number of minutes
$0.2\times60$0.2×60	$=$=	$12''$12′′	Take the decimal component of that number and multiply by $60$60. This is the number of seconds.  Round this to the required decimal places as necessary.
$28.42$28.42°	$=$=	$28^\circ25'12"$28°25′12"	Final answer!

 

Often though you will use a scientific calculator to perform a lot of these conversions.  Make sure you know how to use your calculator correctly.
 

Question 3

Round  $58^\circ28'$58°28′ to the nearest degree.

Think: when rounding I need to identify half way.  For minutes and seconds halfway is at $30$30.  So if the value is more than $30$30, I will round up. If it is less than $30$30 I will round it off.

Do: $0^\circ28'$0°28′, is less than $0^\circ30'$0°30′, so I will round it off. 

$58^\circ28'$58°28′ to the nearest degree is $58^\circ$58°.

 

Question 4

Round $23^\circ16'35"$23°16′35" to the nearest minute.

Think:  When rounding I need to identify halfway.  For minutes and seconds halfway is at $30$30.  So if the value is more than $30$30, I will round up. If it is less than $30$30 I will round it down.

Do: $35$35 seconds is more than $30$30, so I will round up. 

$23^\circ16'35"$23°16′35" to the nearest minute is $23^\circ17'$23°17′.

 

Question 5

Convert $22^\circ8'$22°8′ to seconds.

$22$22 degrees is $22\times60$22×60 minutes = $1320$1320 minutes

Total minutes I have now is $1320+8=1328$1320+8=1328 minutes

$1328$1328 minutes is $1328\times60$1328×60 seconds = $79680$79680 seconds

 

Make sure you know how to use your calculator effectively.  You can use the DMS converter below as well.  

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More Worked Examples

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QUESTION 9

Calculating trigonometric expressions

As with every topic in mathematics that you study there is a conceptual side, (the what you need to know and understand) and the practical side (the what you need to do and answer).  To calculate trigonometric expressions you will need to use a scientific calculator.  Make sure your calculator has the sine (sin), cosine (cos) and tangent  (tan) buttons and that you can enter in degrees, minutes and seconds as angle values.

Examples

Question 1

Evaluate $\tan55^\circ$tan55°

This questions is asking for us to work out the ratio of the opposite and adjacent sides of a triangle with angle $55$55 degrees. Because we don't have the side lengths though, we use the trigonometric function of tangent.  Using your calculator type in $\tan55^\circ$tan55°= and you will get $1.43$1.43 to two decimal places.  

 

Question 2

Find $\theta$θ if

 $\sin\theta=0.65$sinθ=0.65 answer to 2 decimal places

This question is asking us what the angle is if the ratio of the opposite and hypotenuse is 0.65.  To answer this question you use the inverse sin button on your scientific calculator.  Often it looks a bit like this $\sin^{-1}$sin−1

$\sin\theta=0.65$sinθ=0.65

$\sin^{-1}$sin−1 $0.65=40.54$0.65=40.54°

 

Question 3

Evaluate $\frac{\tan35^\circ}{\cos42^\circ-\sin28^\circ}$tan35°cos42°−sin28°​

Think: I will need to remember my order of operations, in this case the subtraction on the denominator comes first, then the division with tan.

Do: $\frac{\tan35^\circ}{\cos42^\circ-\sin28^\circ}=2.56$tan35°cos42°−sin28°​=2.56  

Try to complete the whole question on your calculator in one step, this will reduce rounding errors. 

More Worked Examples

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QUESTION 7