Trignometry
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Triganometry is the study of triangles which connects sides to angles. It uses functions called the sine, cosine and tangent. You can find these on scientific calculators with buttons named sin, cos and tan.
The most common use for these functions is to find sides and angles of a right angled triangle. The rules follow:
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sin A? = opposite/hypotenuse cos A? = adjacent/hypotenuse tan A? = opposite/adjacent |
The sides and angles of a non-right-angled triangle can also be found using the following rules:
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a/sin A =b/sin B = c/sin C a2 = b2 + c2 - 2bc cos A cos A = ( b2 + c2 - a2 ) / ( 2bc ) |
The functions all give graphs which are important. You should know how to sketch them all and know how to use them.
- The Sine Curve
| x? | 0 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | 330 | 360 |
| sin x? | 0.00 | 0.50 | 0.86 | 1.00 | 0.86 | 0.50 | 0.00 | -0.50 | -0.86 | -1.00 | -0.86 | -0.50 | 0.00 |
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This is the curve drawn when you put all the figures from the table above. As you can see, this curve is in a wave form. This wave can continue past 360? and go into the negatives. |
- The Cosine Curve
| x? | 0 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | 330 | 360 |
| cos x? | 1.00 | 0.86 | 0.50 | 0.00 | -0.50 | -0.87 | -1.00 | -0.87 | -0.50 | 0.00 | 0.50 | 0.87 | 1.00 |
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If you look at this curve you can see it is actually the same as the sine curve except it is a different section (i.e. this peaks at 0? where the sine curve peaks at 90?). |
- The Tan Curve
| x? | 0 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | 330 | 360 |
| tan x? | 0.00 | 0.58 | 1.73 |  | -1.73 | -0.58 | 0.00 | 0.58 | 1.73 | | -1.73 | -0.58 | 0.00 |
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The tan curve is very different from the others. It is a non-continuous which breaks as the value at the breaking point ( when x=90 or x=27) is infinity. Again this curve can be continued with the section from x=90 to x=270 repeated. |
From the curves we can see there is always more than one possible value for any number you are working out the inverse ( sin-1 0.5 = 30? or 150? ). The problem is that your calculator only gives you one of the values ( the one below 90? ). You must remember the curves to find the position of the second angle.
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Date: Thu, 14 Sep 2006 Time: 12:00 AM
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GCSE Mathematics revision notes by Richard Tang.
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