Trigonometric Identities |
(Math | Trig | Identities) |
sin(theta) = a / c | csc(theta) = 1 / sin(theta) = c / a |
cos(theta) = b / c | sec(theta) = 1 / cos(theta) = c / b |
tan(theta) = sin(theta) / cos(theta) = a / b | cot(theta) = 1/ tan(theta) = b / a |
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csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
sin^2(x) + cos^2(x) = 1 | tan^2(x) + 1 = sec^2(x) | cot^2(x) + 1 = csc^2(x) |
sin(x y) = sin x cos y cos x sin y | ||
cos(x y) = cos x cosy sin x sin y |
tan(x y) = (tan x tan y) / (1 tan x tan y)
sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)
tan(2x) = 2 tan(x) / (1 - tan^2(x))
sin^2(x) = 1/2 - 1/2 cos(2x)
cos^2(x) = 1/2 + 1/2 cos(2x)
sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )
cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )
angle | 0 | 30 | 45 | 60 | 90 |
---|---|---|---|---|---|
sin^2(a) | 0/4 | 1/4 | 2/4 | 3/4 | 4/4 |
cos^2(a) | 4/4 | 3/4 | 2/4 | 1/4 | 0/4 |
tan^2(a) | 0/4 | 1/3 | 2/2 | 3/1 | 4/0 |
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)
| (Law of Cosines) |
(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)
The z-score
The Standard Normal Distribution
Definition of the Standard Normal Distribution The Standard Normaldistribution follows a normal distribution and has mean 0 and standard deviation 1 |
Notice that the distribution is perfectly symmetric about 0.
If a distribution is normal but not standard, we can convert a value to the Standard normal distribution table by first by finding how many standard deviations away the number is from the mean.
The z-score
The number of standard deviations from the mean is called the z-score and can be found by the formula
x - m
z =
s
Example
Find the z-score corresponding to a raw score of 132 from a normal distribution with mean 100 and standard deviation 15.
Solution