BASIC TRIGONOMETRICAL IDENTITIES 

Geometry  
1. TRIGONOMETRICAL IDENTITIES  
Think back to the oneunit hypotenuse triangle. If we apply Pythagoras' theorem we obtain the most important basic trigonometrical identity: 
Use this identity together with the one in the box below, which relates the tangent to the sine and cosine of an angle, to work out all the other standard trigonometrical identities when you know one of them. 
1. Prove
that the above trigonometrical identities are true for different values
of angle A.
2. Work out the cosine and tangent of an angle given that the sine of the same angle is 1/3. 3. Now work out the sine and tangent if the cosine of the angle is 4. How do we work out the sine and cosine of an angle when the tangent of the angle is 1? 
2. trigonometrical ratios of complementary angles  
Given that the angles in any triangle add up to 180º, the two acute angles in a rightangled triangle always add up to 90º. Angles that add up to 90º are also known as complementary angles. From now on the Descartes program will give the trigonometrical ratios for angles A and C simultaneously, as we now know that they are complementary angles. 

5. Can you see a connection between the trigonometrical ratios of angles A and C? 6. What about when angle A is: 45º, 60º, 33.4º, 72º and 85.7º. Are the relations above between the ratios of angles A and C maintained? 7. Sin A = 0.391 and cos A = 0.921. Find tan A and the three trigonometrical ratios of the complementary angle of angle A.

Miguel García Reyes  
Descartes Project. Year 2001  
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