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Question:
When you say below find
the exact value of each of the remaining trigonometric functions of `theta.
Do you mean find all the fundamental identaties etc.
Becaues x< 0 ..I've determined x=(-18/13) by
using the pythagorean identaty of the circle. So, it's merely a matter of working out the
other four identaties. But I want to understand the question fully in case similar
language is used in a test...
Question # 36..... 5.3 section
find the exact value of each of the remaining
trigonometric functions of `theta.
36. sin `theta = -5/13, `theta in quadrant 3
Response:
On a unit circle we know that at angular position theta we have y = -5/13, and that
the hypotenuse is 1. Thus x^2 + y^2 = 1.
So x = +- sqrt(1 – y^2) = +- sqrt(1 – (5/13)^2 ) = +- sqrt(1 – 25 /
169) = +- sqrt(169 / 169 - 25 / 169) = +- sqrt(144 / 169) = +-12/13.
Since theta is in the third quadrant we know that x is negative. We therefore
conclude that x = -12 / 13.
Now we know that x = -12/13, y = -5/13 and r = 1.
Using the circular definitions of the trigonometric functions we proceed to write down
their values:
Cos(theta) = x / r = -12/ 13
Tan(theta) = y / x = (-5/13) / (-12/13) = 5/12
Cosec(theta) = r / x = 1 / (-12/13) = -13/12
Etc.
New Page 1
Question:
When you say below find
the exact value of each of the remaining trigonometric functions of `theta.
Do you mean find all the fundamental identaties etc.
Becaues x< 0 ..I've determined x=(-18/13) by
using the pythagorean identaty of the circle. So, it's merely a matter of working out the
other four identaties. But I want to understand the question fully in case similar
language is used in a test...
Question # 36..... 5.3 section
find the exact value of each of the remaining
trigonometric functions of `theta.
36. sin `theta = -5/13, `theta in quadrant 3
Response:
On a unit circle we know that at angular position theta we have y = -5/13, and that
the hypotenuse is 1. Thus x^2 + y^2 = 1.
So x = +- sqrt(1 – y^2) = +- sqrt(1 – (5/13)^2 ) = +- sqrt(1 – 25 /
169) = +- sqrt(169 / 169 - 25 / 169) = +- sqrt(144 / 169) = +-12/13.
Since theta is in the third quadrant we know that x is negative. We therefore
conclude that x = -12 / 13.
Now we know that x = -12/13, y = -5/13 and r = 1.
Using the circular definitions of the trigonometric functions we proceed to write down
their values:
Cos(theta) = x / r = -12/ 13
Tan(theta) = y / x = (-5/13) / (-12/13) = 5/12
Cosec(theta) = r / x = 1 / (-12/13) = -13/12
Etc.