Trigonometric Identities Sin, Cos and Tan

In my experience the best way to differentiate teaching the sin, cos and tan trigonometric identities is through discovery.   

I begin the lesson by explaining we are going to discover some relationships between the three sides of a right-angled triangle and the interior angles.  We discuss theta (θ) as the name of the angle we will be working with and I explain the adjacent and opposite sides of the triangle depend on the position of theta.  The hypotenuse side is opposite the right-angle and is always the longest side.

Introducing Trigonometric Identities
Introducing Trigonometric Identities

Introduction to Trigonometry

To begin the discovery, I ask students to draw a right-angled triangle where the adjacent is 5 cm and theta is 30°.   Students then measure the length of the opposite and hypotenuse sides in millimetres.  After checking their progress, I feedback by asking students to share their measurements of the two sides.  This helps to identify mistakes early on.

Introducing Trigonometric Identities

To save time I give the students a handout the table above to stick in their books.  Next, I ask the class to enter their results for the 30° triangle in the orange rows.  I demonstrate how to calculate the values for the blue and green rows for the same triangle using a calculator.  All these results are entered in the table. 

Students work independently for about the next 15 minutes to construct the triangles.  They measure the sides of each triangle and calculate the necessary values.  I remind everyone to enter the results in the table as they go.

Discovering Sin, Cos, Tan

When most of the class have got halfway through completing the table, I ask students to take a minute and try to identify any patterns within their results.  At this stage I do not provide any further prompts, so the task is as open as possible. I ask students to share any patterns they have noticed with the person next to them.

As students continue with the triangles, I encourage them to consider whether future values fall in line with their pattern or contradict it.  If future results contradict their patter I encourage students to reassess.

For those who have not found a pattern I encourage them to check their measurements and whether the data is recorded correctly in the table.

Trigonometric Identities

Students work at different paces, so I encourage peer support.  For instance, one student records the results when theta is 30°, 50° and 70° and the other records the results for remaining angles.  Either way, every student is expected to have a completed set of results in their exercise book.

About 35 minutes into the lesson most students have completed the table of results.  I now stop the class and ask them to discuss with their peers any patterns they have noticed.

After their discussion I ask students to present their patterns in the form of equations to me on mini whiteboards.  About three quarters of the class present their patterns as sinθ = opp/hyp, cosθ = adj/hyp and tanθ = opp/adj.  The remaining quarter describe the same relationships in words.

Applying Trigonometric Identities to Solve Problems

As part of an extended plenary I demonstrate how to use the three equations to calculate unknown lengths in these right-angled triangles.  You can watch a video of this demonstration here.

Typically, I work through questions a and b then ask students to attempt c and d in their books before I feedback.

In the final part of the lesson I ask the students to solve this problem on mini whiteboards.  They are free to work on their own or with a peer for support.  Again, having checked their progress I feedback on the main board.

Future lessons on Trigonometric Identities

In the next lesson, students learn more about using trigonometric identities to calculate unknown angles in a triangle before moving on to lengths.  You can view the medium term plan here.


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About Mr Mathematics

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.


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