Trigonometry Summary
- cos(a) = ADJ/HYP
- sin(a) = OPP/HYP
- The law of sines states that in any triangle, the ratio of the sine of an angle to the opposite side length is equal for all three angles and sides.
- sin(A)/a = sin(B)/b = sin(C)/c
- Where A, B, C are the angle measures of the triangle, and a, b, and c, are the opposite side lengths.
- The law of cosines states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the opposite angle.
- sec(a) = 1/cos(a)
- csc(a) = 1/sin(a)
- cot(a) = 1/tan(a)
- tan(a) = sin(a)/cos(a)
- cot(a) = cos(a)/sin(a)
$$sin^2(a) + cos^2(a) = 1$$ $$tan^2(a) + 1 = sec^2(a)$$ $$cot^2(a) + 1 = csc^2(a)$$
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sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
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sin(a-c)=sin(a)cos(c)-sin(c)cos(a)
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cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
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cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
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tan(x+y)=(tan(x) + tan(y))/(1-tan(x)tan(y)).
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tan(x-y)=(tan(x) - tan(y))/(1+tan(x)tan(y)).
$$sin(2a)=2sin(a)cos(a)$$ $$cos(2a)=cos^2(a)-sin^2(a)$$ $$tan(2a)=2tan(a)/(1-tan^2(a))$$
- sin(a/2) = +- sqrt((1-cos(a)/2))
- cos(a/2) = +- sqrt((1+cos(a)/2))
- tan(a/2) = +- sqrt((1-cos(a))/(1+cos(a))) = (1-cos(a))/sin(a)=sin(a)/(1+cos(a))
- sin(-a) = -sin(a)
- cos(-a) = cos(a)
- tan(-a) = -tan(a)
- sin(a+2pi) = sin(a)
- cos(a+2pi) = cos(a)
- tan(a+pi) = tan(a)
- sin(a) = cos(pi/2-a)
- cos(a) = sin(pi/2-a)
- tan(a) = cot(pi/2 - a)
- cot(a) = tan(pi/2 - a)
- sec(a) = csc(pi/2-a)
- csc(a) = sec(pi/2-a)
$$2*cos^2(a)-1=cos(2a)$$ $$cos^2(a)=(1/2) * (1+cos(2a))$$ $$cos^2(a)+sin^2(a)=1$$ $$cos(2a)=1-2sin^2(a)$$ $$sin^2(a)=(1/2)*(1-cos(2a))$$
- The word Sine originates from the Latin word Sinus, and it means curve or bay. Here it is used to mean: The line formed by the inside triangle, that is closest to the curve.
- The word Secant, is from the Latin word secare, and it means to cut. Here it is used to mean: The line formed by the inside triangle, that when extended, cuts the curve.
- The word Tangent, in Latin, means touching. Here it is used to mean: The line that touches the curve, at the same point where the inside triangle touches the curve.
- The "co-" prefix comes from Latin, it literally means "complimentary", or as I like to think of it: "at 90 degrees to something"
- Cosine for: The line complementary to the Sine. Or stated another way: The line at 90 degrees to the line of the triangle, closest to the curve.
- Cosecant for: The line complementary to the Secant. Or rather: The line at 90 degrees to Secant that, when extended, also cuts the curve.
- Cotangent for: The line complementary to the Tangent. Or perhaps better understood as: The line-segment of the extended touching-line.