Re: trigonometric functions


"conrad" <conrad@xxxxxxxxxx> wrote in message
news:1171603393.176058.25450@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Feb 15, 9:30 pm, "Michael" <mchlg...@xxxxxxx> wrote:
On Feb 15, 7:10 pm, "conrad" <con...@xxxxxxxxxx> wrote:

http://www.bbc.co.uk/education/asguru/maths/13pure/04trigonometry/21f...
Given this image, what is the
relationship between the right triangle
and 120 degrees? I thought the trig
functions were defined in terms of
right triangles. But how can they be applied
to finding angles that lie outside of right
triangles as in the above image?

--
conrad

It sounds like you're familiar with definitions such as sine =
opposite / hypotenuse.

If you work it out, those definitions only exist for angles <= 90
degrees, and sine, cosine, tangent, etc. will always be positive.

What's shown in the picture is how one extends the trig functions to
all angles (at least up to 360 degrees). Note that the angle shown is
120 degrees, which is > 90, and that the x coordinate is negative, not
positive.

With a little geometry, you can work out sine(120) and cosine(120) in
terms of sine(60) and cosine(60) (with minus signs where appropriate).

Then you can work out the signs for the other two quadrants.

And suddenly you have the trig functions defined for all angles.

Michael

But in the image, for point P, how does knowing
say the length of the opposite side
and the length of the hypotenuse, tell me
that the angle is 120 and not the angle created
by the right triangle in terms of point P?

AS Michael had said the opposite/hypotenuse definition only applies to
right-angle triangle (with angles less than 90 degrees).

If instead you imagine the angle (between 0 and 360 degrees) being that
swept by a point (x,y) being swept by the radius of a circle (the technical
term is the locus).

Then the tangent of the angle from the x-axis is x/y
sine is y/r and cosine is x/r

This then determines the sign of the trig function within each quadrant

Angle Sign of x Sign of y Sign
of tan Sign of sine Sign of cos

0-90 + +
+ + +
90-180 -
+ -
-
180-270 - -
+ - -
270-360
- - -
+

Alternatively you can imagine the angle of 120 degrees in the upper-left
quadrant being part of a right-angled triangle in that quadrant with the
adjacent (x value) being negative and the opposite (y value) being positive.

Likewise the sign of the sides of the triangle as the angle sweeps round
through the 360 degrees.

Nick


.

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