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--- urp:trig [2021-10-12]
nerf_herder created
+++ urp:trig [2021-10-21]
nerf_herder
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 transformations of sine or cosine function:
- y = a * sin(k(x + c)) + d
+ y = a * sin(k(x - c)) + d
-a = amplitude, -c = phase shift, d = vertical shift, 2pi/k = period
+{{:urp:sinewave.jpg?418|Sine Wave Transformations}}
-for tan, the period is pi/k
+(Graphics from www.Desmos.com, labeling was added) a = amplitude, c = phase shift, d = vertical shift, 2π/k = period
+for tan, the period is π/k
+----------
+double angle: 2θ
+  sin(2θ) = 2sin(θ)cos(θ)
+  cos(2θ) = cos²(θ) - sin²(θ)
+          = 2*cos²(θ) - 1
+          = 1 - 2*sin²(θ)
-double angle: 2theta
-  sin(2theta) = 2sin(theta)cos(theta)
-  cos(2theta) = cos^2(theta) - sin^2(theta)
-              = 2*cos^2(theta) - 1
-              = 1 - 2*sin^2(theta)
 negative angles:
-  sin(-theta) = -sin(theta)
+  sin(-θ) = -sin(θ)
-  cos(-theta) = cos(theta)  => tan(-theta) = -tan(theta)
+  cos(-θ) = cos(θ)
+  tan(-θ) = -tan(θ)
 additive, subtractive:
- sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
+  sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
+  cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
- sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
+  sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
- cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
+  cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
 complements:
- sin(theta) = cos(pi/2 - theta)
+  sin(θ) = cos(π/2 - θ)
- cos(theta) = sin(pi/2 - theta)
+  cos(θ) = sin(π/2 - θ)
+=== See also ===
+  * https://www2.clarku.edu/faculty/djoyce/trig/identities.html
+  * 11 pages of definitions, postulates and theorems: http://www.ouchihs.org/ourpages/auto/2013/7/26/52822673/Geo-PostulatesTheorems-List.pdf
+Back to [[math]] page.

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