Trigonometric Functions

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Move the point labeled "Move" to adjust the angle between the ray and the x-axis.

If Hypoteneuse = r = 1 (a unit circle), then:

Opposite = sin(θ) = y

Adjacent = cos(θ) = x

SOHCAHTOA:

SOH: Opposite / Hypoteneuse = sin(θ) = y / r

CAH: Adjacent / Hypoteneuse = cos(θ) = x / r

TOA: Opposite / Adjacent = tan(θ) = y / x

Other Functions:

Hypoteneuse / Opposite = csc(θ) = r / y = 1 / sin(θ)

Hypoteneuse / Adjacent = sec(θ) = r / x = 1 / cos(θ)

Adjacent / Opposite = cot(θ) = x / y = 1 / tan(θ)

1 = csc(θ)^2 - cot(θ)^2

1 = sec(θ)^2 - tan(θ)^2

1 = sin(θ)^2 + cos(θ)^2

1 = sec(θ) - exsec(θ)

1 = csc(θ) - coexsec(θ)

1 = vers(θ) + cos(θ)

1 = sin(θ) + covers(θ)

hav(θ) = vers(θ) / 2

Pythagorean Theorem:

c^2 = a^2 + b^2

where Adjacent = a, Opposite = b, and Hypoteneuse = c.

Law of Sines:

2 * r = a / sin(A) = b / sin(B) = c / sin(C)

where "r" is the radius of the circumcircle,

and a = Adjacent, b = Opposite, and c = Hypoteneuse,

and A = the angle opposite a, B = the angle opposite b, and C = the angle opposite c.

Law of Cosines:

cos(A) = (c^2 + b^2 - a^2) / (2 * b * c)

where a = Adjacent, b = Opposite, and c = Hypoteneuse,

and A = the angle opposite a.

Law of Tangents:

(a + b) / (a - b) = tan((A + B) / 2) / tan((A - B) / 2)

where a = Adjacent, b = Opposite, and c = Hypoteneuse,

and A = the angle opposite a, and B = the angle opposite b.

Dot Product:

A•B = cos(θ) = x_A * x_B + y_A * y_B

where A and B are vectors with lengths equal to 1.

Download the worksheet here. Download an older Geometer's Sketchpad version of this worksheet here.

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This page © Copyright 2009 Michael Horvath. Last modified: February 21 2010 20:38:37.