Using the Calculator in Trigonometry Revision Notes for HSC SSCE Mathematics Standard

Using the Calculator in Trigonometry

Understanding angle measurement in trigonometry

When working with trigonometry, we measure angles using three units: degrees, minutes, and seconds. This system is similar to how we measure time, which makes it easier to remember.

chatImportant

Before using your calculator for trigonometric calculations, you must ensure it is set to degree mode. This is essential for getting correct answers in your SSCE course.

The conversion relationships are:

$1° = 60′$ (one degree equals sixty minutes)
$1′ = 60″$ (one minute equals sixty seconds)

These conversions work just like time: there are 60 minutes in one degree, and 60 seconds in one minute.

Finding trigonometric ratios

When you need to calculate the value of a trigonometric ratio (such as $\sin 60°$ or $\cos 40.5°$ ), you can use your calculator to find the answer quickly and accurately.

Calculator steps for finding ratios

To find a trigonometric ratio:

Make sure your calculator is in degree mode
Press the appropriate trigonometric key ( $\sin$ , $\cos$ , or $\tan$ )
Enter the angle measurement
For angles with minutes and seconds, use the $°′″$ or DMS button
Press equals or exe to get the result

Worked examples

Let's look at some examples of how to find trigonometric ratios using a calculator.

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Worked Example: Finding $\sin 60°$

Press: $\sin$ | 60 | $=$ or exe
Calculator shows: $0.8660254038$
Rounded to two decimal places: $0.87$

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Worked Example: Finding $2\cos 40.5°$

Press: 2 | $\cos$ | 40.5 | $=$ or exe
Calculator shows: $1.52081131$
Rounded to two decimal places: $1.52$

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Worked Example: Finding $\frac{3}{\tan 75°}$

Press: 3 | ÷ | $\tan$ | 75 | $=$ or exe
Calculator shows: $0.8038475773$
Rounded to two decimal places: $0.80$

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Worked Example: Finding $\sin 34°20′$

Press: $\sin$ | 34 | $°′″$ | 20 | $°′″$ | $=$ or exe
Calculator shows: $0.5640065581$
Rounded to two decimal places: $0.56$

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Notice that when entering angles with minutes (like $34°20′$ ), you need to use the special $°′″$ or DMS button on your calculator. Some calculators may require you to select this from an options menu.

Finding angles from trigonometric ratios

Sometimes you know the value of a trigonometric ratio and need to find the angle. For this, you use the inverse trigonometric functions: $\sin^{-1}$ , $\cos^{-1}$ , and $\tan^{-1}$ .

Calculator steps for finding angles

To find an angle from a trigonometric ratio:

Ensure your calculator is in degree mode
Press the SHIFT key (or 2nd function key)
Press the appropriate inverse trigonometric key ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ )
Enter the ratio value
Press equals or exe
If needed, use the $°′″$ or DMS button to convert the answer to degrees, minutes, and seconds

Worked examples

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Worked Example: Finding $\theta$ when $\sin \theta = 0.6123$

Press: SHIFT | $\sin^{-1}$ | 0.6123 | $=$ or exe
Calculator shows: $37.75599438$
Rounded to nearest degree: $\theta \approx 38°$

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Worked Example: Finding $\theta$ when $\tan \theta = 1.45$

Press: SHIFT | $\tan^{-1}$ | 1.45 | $=$ or exe
Calculator shows: $55.409396...$
Convert to minutes using $°′″$ or DMS: $55°24′27.76″$
Rounded to nearest minute: $\theta \approx 55°24′$

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Worked Example: Finding $\theta$ when $\sin \theta = \frac{4}{5}$

Press: SHIFT | $\sin^{-1}$ | $\frac{4}{5}$ | $=$ or exe
Calculator shows: $53.130102...$
Rounded to nearest degree: $\theta \approx 53°$

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Worked Example: Finding $\theta$ when $\cos \theta = \frac{1}{\sqrt{2}}$

Press: SHIFT | $\cos^{-1}$ | $\frac{1}{\sqrt{2}}$ | $=$ or exe
Calculator shows: $45°$
Answer: $\theta = 45°$

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Exam tip: When finding angles, your calculator will give you the answer in decimal degrees. If the question asks for the answer in degrees and minutes, remember to use the DMS conversion button to change the format. Make sure you round to the correct number of decimal places or the nearest degree/minute as specified in the question.

Remember!

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Key Points to Remember:

Always check your calculator is in degree mode before starting trigonometric calculations
Remember the conversion: $1° = 60′$ and $1′ = 60″$
For finding ratios: Press the trig key ( $\sin$ , $\cos$ , $\tan$ ) first, then enter the angle
For finding angles: Use SHIFT plus the inverse trig key ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ )
Use the $°′″$ or DMS button to enter or convert angles in minutes and seconds
Always round your final answer to the number of decimal places or accuracy requested in the question

Using the Calculator in Trigonometry (HSC SSCE Mathematics Standard): Revision Notes