Some important definitions:
Addition formulae
[pic]
[pic]
[pic] |Special angles
[pic]
[pic]
[pic] |Double angle formulae (learn!)
[pic]
[pic]
[pic]
Also learn these rearrangements:
[pic]
[pic] | |
|Solving equations: | |Solving equation example: |
|To solve equations [pic], get one value from your calculator and then | |Solve [pic] for [pic] |
|the 2nd value is | | |
|[pic]. | |Solution: [pic] (def’n of cotx) |
|Further values can be obtained by adding (or taking away) multiples of |[pic] |[pic] (multiply by sinx) |
|2π or 360°. | |We now use the relationship[pic]to get equation in terms of cosx only. |
| | |[pic] |
| | |[pic] |
| | |Using the quadratic formula: |
| | |[pic] |
| | |So, x = 0.675 or x = 2π – 0.675 = 5.61. |
|Expressions for [pic] |Step 3: Find R (by eliminating α). |We can now see that the maximum and minimum values of 2cosx – 3sinx will |
| |(1)2 + (2)2: [pic] |be [pic] and [pic] respectively. |
|Example: Write 2cosx – 3sinx in the form [pic] . |[pic] | |
| | |The maximum value occurs when |
|Step 1: Write out the expansion of [pic]: |Step 4: Find α (by eliminating R): |[pic] |
|[pic] |(2)/(1): [pic] |So: x + 0.983 = 0 or 2π |
| | |i.e. x = -0.983 or 5.30. |
|Step 2: Compare expressions: |Step 5: So, 2cosx – 3sinx = [pic] | |
|[pic] | | |
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