# General Angle Trigonometry Investigation

*By toni On 27 September, 2017 · Leave a Comment*

DO NOT USE CALCULATORS FOR THIS WHOLE CLASS ACTIVITY.

**YOU NEED TO KNOW: t****hat the cosine and sine functions for ALL ANGLES are given by the coordinates of points on the UNIT CIRCLE. The x-coordinates of points on the unit circle give the cosine of the angle between the x-axis and the radius measured anti-clockwise and the y-coordinates give the sine of the angle.**

In this activity learners will share the work of taking readings of the coordinates of points on the unit circle for the angles 10^o, 20^o , ..., 350^o, 360^o. Then you will plot these points to discover what the graphs of the cosine and sine functions look like.

You will need a ruler and protractor and Sheet 1 – the Unit Circle, and one card for recording readings for each pair of learners, and Sheet 2 for recording the results from the whole class, and Sheet 3 – Graph for plotting readings from Sheets 1 and 2.

**STEP 1** On Sheet 1 use a protractor to measure an angle of 60^{o }anti-clockwise from the x-axis and draw the lines OP and PNQ. The right angled triangle OPN has edges OP=1, ON=cos60^{o} and PN=sin60^{o}. From the lines on the grid you should see clear steps of 0,1 from -1 to + 1 in both x and y directions so you can take readings to 2 decimal places. Read the coordinates of P from the grid.

For 300^{o} the radius is OQ and the readings are (cos300^{o}; sin300^{o}) =(0,5;-0,87) *Note the sign. *Readings should be recorded in a table like this.

**STEP 2 **Draw accurately the 6 angles on your card in this way, then take readings from the grid of the x-coordinates for cosines and the y-coordinates for sines correct to 2 decimal places, and to fill in the tables on your cards.

**STEP 3 As a class **fill in Sheet 2 using the results from the whole class. Several learners will have taken each reading so you can check that the readings are correct.

**STEP 4** The next step is to plot the points (A, cos A) using Output 1 from Sheet 2. ONLY THE FIRST TWO COLUMNS ARE USED FOR cos A. You could cover the Output 2 column for sinA or fold sheet 2 to hide that column.

On Sheet 3 plot **on the horizontal axis** the values of angle A in degrees from the 1^{st} column and **on the vertical axis** the values of cos A from the 2^{nd} column.

**This graph of f(A)= cosA **should be a** NICE ROUNDED CURVE.**

**STEP 5 **Plot the points (A, sinA) of the function g(A)= sin A on Sheet 3 using Output 2 from sheet 2. Plot on the horizontal axis the values of angle A in degrees from the 1^{st} column and, on the vertical axis, the values of sin A from the 3^{rd} Again the graph should be a **NICE ROUNDED CURVE.**

**STEP 6 **What do you notice when you compare the graphs of f(A) = cos A and g(A) = sin A?

What do you think happens if you go on to 370^{o}, 380^{o}, and so on and on ?

What happens for negative angles?

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