By the end of this module you will be expected to have learnt the following formulae:
If you have a polynomial f(x) divided by x - c, the remainder is equal to f(c). Note if the equation is x + c then you need to negate c: f(-c).
A polynomial f(x) has a factor x - c if and only if f(c) = 0. Note if the equation is x + c then you need to negate c: f(-c).
![{\displaystyle b^{x}b^{y}=b^{x+y}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8c2641be6fdf034073959ab8768efa78781e39)
![{\displaystyle {\frac {b^{x}}{b^{y}}}=b^{x-y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d49193923b42f404c639c849c2941d8f7adce1)
![{\displaystyle \left(b^{x}\right)^{y}=b^{xy}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d7eb5cc68693c502d8e0b737e609fe1b48c404)
![{\displaystyle a^{n}b^{n}=\left(ab\right)^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17dc0c248c24fa4dc3c1627b60f05eea108e1f4e)
![{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39b91d09e405cbb252de7c9c1ffe632d3a350d9f)
![{\displaystyle b^{-n}={\frac {1}{b^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb09df7fe703ced09c5c135a5cb86c6e94ea77b)
where c is a constant
![{\displaystyle b^{1}=b\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4873e8f9e3c7d9444b4a95a5d920b6db29c1a40d)
![{\displaystyle b^{0}=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5746fd6092358a351aa813e9a4a60ac2a8a7a80)
The inverse of
is
which is equivalent to
Change of Base Rule:
can be written as
When X and Y are positive.
![{\displaystyle \log _{b}XY=\log _{b}X+\log _{b}Y\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9f7e2a4ca6f973cf3bc879f287b346e184be7b)
![{\displaystyle \log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03df55fc1f6e3d73aadc13503f95964e550716fd)
![{\displaystyle \log _{b}X^{k}=k\log _{b}X\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1fb60ceb5f3f6636276e62924774b2b04fa440)
where X is the degree, y is the minutes, and z is the seconds.
Note: θ need to be in radians
Note: θ need to be in radians.
Function |
Written |
Defined |
Inverse Function |
Written |
Equivalent to
|
Cosine
|
![{\displaystyle \cos \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4082f787030c0c48e20d9270415c9e6208aa6b07) |
![{\displaystyle {\frac {Adjacent}{Hypotenuse}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6bf301ff61c0bf2afcfcfd04c81e6d29b86b1f) |
![{\displaystyle \arccos \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52fcb75706c3a5e20721756bb57cf1f668a27dfb) |
![{\displaystyle \cos ^{-1}\theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f466d5033d527fc4d36f9c798d2b5485c939bb) |
|
Sine
|
![{\displaystyle \sin \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d36af63acb13b05b295b62c463015069473f44f) |
![{\displaystyle {\frac {Opposite}{Hypotenuse}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f53aab7509cc322a232f6001f4982ecb34891c) |
![{\displaystyle \arcsin \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1080e048c5ec9e0ea7be85fcb2e842157070938) |
![{\displaystyle \sin ^{-1}\theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14fd782d8c8a35b02f8b88970ade0618c3e354a3) |
|
Tangent
|
![{\displaystyle \tan \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42bbd1b0f25d35d756e1a2f1ed63ba54141cba91) |
![{\displaystyle {\frac {Opposite}{Adjacent}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1cabbf82e44d75966e412a72c0d6eb8f078900) |
![{\displaystyle \arctan \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9b28dfa9159654da014c4b6d43e884aa21fd00) |
![{\displaystyle \tan ^{-1}\theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cea66d8054ef0baa831cf48cb02ba623207158) |
|
You need to have these values memorized.
![{\displaystyle \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/228647b7d4a18b6c8c0c390b439a61da8fafec76) |
![{\displaystyle rad\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecfb4e44464e34b4c6794bda097d2356f0383a1) |
![{\displaystyle \sin \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d36af63acb13b05b295b62c463015069473f44f) |
![{\displaystyle \cos \theta \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4082f787030c0c48e20d9270415c9e6208aa6b07) |
|
![{\displaystyle 0^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef) |
0 |
0 |
1 |
0
|
![{\displaystyle 30^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f29df9c101d2a8dae3f1552342cfe4c3adb76c) |
![{\displaystyle {\frac {\pi }{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da430867901fd359c000b52f2bd70b36cf5e2182) |
![{\displaystyle {\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55) |
![{\displaystyle {\frac {\sqrt {3}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4864a0c173339d1d88e89ca3c943f016744c879a) |
|
![{\displaystyle 45^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28223ddedeb94a84bb15474cc64b5ce436cbe50) |
![{\displaystyle {\frac {\pi }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de) |
![{\displaystyle {\frac {\sqrt {2}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb9b5960bf5eae3065db9c23495e465f5fef61e) |
![{\displaystyle {\frac {\sqrt {2}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb9b5960bf5eae3065db9c23495e465f5fef61e) |
|
![{\displaystyle 60^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c42292485b447b7f627a7accd90d5b439c11d0) |
![{\displaystyle {\frac {\pi }{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c83c684a603005cda4feb8eea0254143ffb0e16) |
![{\displaystyle {\frac {\sqrt {3}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4864a0c173339d1d88e89ca3c943f016744c879a) |
![{\displaystyle {\frac {\sqrt {1}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9ba7de6e29f6e46447d996ac8efd4827123533) |
|
![{\displaystyle 90^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6) |
![{\displaystyle {\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2) |
1 |
0 |
-
|
The reason that we add a + C when we compute the integral is because the derivative of a constant is zero, therefore we have an unknown constant when we compute the integral.
, F is the anti derivative of f such that F' = f
![{\displaystyle \int _{a}^{b}f\left(x\right)\ dx=-\int _{b}^{a}f\left(x\right)\ dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26fc1ea6cad860d9a49f5c552cdd6a91f18c7281)
![{\displaystyle \int _{a}^{a}f\left(x\right)\ dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f06872d083ab5114be2cf9f94910e50fa23b1a)
- Area between a curve and the x-axis is
![{\displaystyle \int _{a}^{b}y\,dx\ ({\mbox{for}}\ y\geq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6ee8e35d0979445e96e0f4156e339e722c92e2)
- Area between a curve and the y-axis is
![{\displaystyle \int _{a}^{b}x\,dy\ ({\mbox{for}}\ x\geq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b719319264eee2a047435eb5e6ff142a2dac4172)
- Area between curves is
![{\displaystyle \int _{a}^{b}{\begin{vmatrix}f\left(x\right)-g\left(x\right)\end{vmatrix}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f33f743a36bce8f1d25b08e0479c4336e1b403)
Where:
Where:
n is the number of strips.
and