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Unformatted text preview: 1.3: The xyr Definitions of the Trigonometric Functions
The six trigonometric functions can be defined in several ways. The first way may be referred to as the xyr
definitions, where x and y are the coordinates of a point on the terminal side of the angle when placed in standard
position, and r is the distance of that point from the origin. The inputs to the functions will be angles, and the
outputs will be ratios of the numbers x, y, and z.
Place θ in standard position, pick any point on the terminal side of θ except the origin itself, and use the coordinates of this
point to compute its distance from the origin. r = x2 + y 2 , r > 0
We define the six trigonometric functions as follows.
(sine, cosine, tangent, cotangent, secant, cosecant)
Ex: Given that P = (3, 5) , compute values sin θ = for the six trigonometric functions of the angle
in standard position. First we compute r: y
r cos θ = x
r r = 32 + 5 2
tan θ = r = 9 + 25 y , x≠0 x r = 34 cot θ = x ,y≠0 y
sec θ =
We can now write down these ratios, given
that we already know that r , x≠0 x x = 3 and y = 5 . csc θ = r ,y≠0 y
5 sin θ = = 34 34
3 cos θ = 34
tan θ = = 3 34
34 The top two ratios on the left have been multiplied by in
order to rationalize the denominators. While your textbook
always does this when listing answers, it is not important to me
whether you leave the radical in the denominator or not. Note that if the angle θ lies in one of the quadrants (as opposed
to being quadrantal), then neither x nor y will equal zero, and all 3 six trigonometric functions will be defined for 5 quadrantal, then either sec θ = 34
3 cs c θ = 34 5
3 c ot θ = 5 34 34
5 θ . If θ is x = 0 or y = 0 , and two of these functions will be undefined. ...
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This note was uploaded on 11/16/2011 for the course MAC 1114 taught by Professor Cohen during the Fall '09 term at Santa Fe College.
- Fall '09