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If Then Trigonometry

Finding the exact values of sine, cosine and tangent of angles if given a different trig ratio.

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Solve these "If Then" questions without using a calculator but giving exact answers in their simplest form. Use the / symbol to show a fraction and the root button to insert the square root sign if required.

If \( \sin \theta = \frac{3}{5} \)
then find \( \cos \theta \)

Correct Wrong

If \( \cos \theta = \frac{12}{13} \)
then find \( \tan \theta \)

Correct Wrong

If \( \tan \theta = \frac{3}{4} \)
then find \( \sin \theta \)

Correct Wrong

If \( 13\cos \theta = 5 \)
then find \( \sin \theta \)

Correct Wrong

If \( 5\sin \theta = 4 \)
then find \( \tan \theta \)

Correct Wrong

If \( 3\tan \theta = 4 \)
then find \( \cos \theta \)

Correct Wrong

If \( \tan \theta = 1 \)
then find \( \sin \theta \)

Correct Wrong

If \( 2\cos \theta = 1 \)
then find \( \tan \theta \)

Correct Wrong

If \( 2\sin \theta = \sqrt{3} \)
then find \( \cos \theta \)

Correct Wrong



Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

Why am I learning this?

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

Comment recorded on the 17 June 'Starter of the Day' page by Mr Hall, Light Hall School, Solihull:

"Dear Transum,

I love you website I use it every maths lesson I have with every year group! I don't know were I would turn to with out you!"

Comment recorded on the 14 September 'Starter of the Day' page by Trish Bailey, Kingstone School:

"This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils love it!

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Featured Activity



Interactive, randomly-generated, number-based logic puzzle based on the Latin square designed to develop numeracy skills. These puzzles are drag and drop and can earn you a Transum Trophy.


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Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.


If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

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When planning to use technology in your lesson always have a plan B!

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Description of Levels



Surds - Make sure you understand what surds are before starting the levels below.

Common Trig Ratios Level 1 - Find exact trig values for special angles up to and including ninety degrees

Common Trig Ratios Level 2 - Find the indicated lengths by solving trigonometric questions with exact solutions

Common Trig Ratios Level 3 - Mixed questions on exact trig values of special angles up to and including ninety degrees

Common Trig Ratios Level 4 - Find exact trig values for angles between ninety and three hundred and sixty degrees

Common Trig Ratios Level 5 - Solving trigonometric equations with given domains

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Trigonometry including visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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The questions in this exercise are designed to be solved by drawing a diagram of a right-angled triangle, choosing the lenghts of two of the sides using the given ratio then use Pythagoras' theorem to figure out the length of the third side. The required trig ratio can then be found from the diagram.

For example, if \(\tan \theta = \frac{8}{15} \) then find \( \sin \theta \)

Firstly sketch a righ-angled triangle containing the angle \( \theta \), opposite 8 and adjacent 15.


The length of the hypotenuse can be calculated using pythagoras' Theorem to be \( \sqrt{8^2 + 15^2} = 17\).

Finally \( \sin \theta \) can be calculated as the opposite over the hypotenuse which is \( \frac{8}{17} \).

Common Trigonometric Ratios Video

Helpful Diagrams

Common Trig Ratios

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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