SOLUTION: find sin theta and cos theta if cot theta =7/5 and sec theta <0

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Question 1205832: find sin theta and cos theta if cot theta =7/5 and sec theta <0
Found 4 solutions by MathLover1, Theo, MathTherapy, math_tutor2020:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

find sin+%28theta%29 and cos%28+theta+%29 if cot%28+theta+%29=7%2F5+and sec%28+theta%29+%3C0

cot%28+theta+%29=cos%28+theta+%29%2Fsin+%28theta%29
cos%28+theta+%29%2Fsin+%28theta%29=7%2F5
%28+b%2Fc+%29%2F+%28a%2Fc%29=7%2F5
b%2Fa+=7%2F5 => a=5,+b=7

c^2=a^2+b^2
c=sqrt%285%5E2%2B7%5E2%29
c=sqrt%2874%29
or
c=-sqrt%2874%29

sin+%28theta%29=5%2Fsqrt%2874%29 or+sin+%28theta%29=-5%2Fsqrt%2874%29+
cos%28+theta+%29=7%2Fsqrt%2874%29 or cos+%28theta%29=-7%2Fsqrt%2874%29+
since given
cot%28+theta+%29%3E0 and sec%28theta%29%3C0+, we use negative value for +sin+%28theta%29 and cos%28theta%29+

so,
sin+%28theta%29=-5%2Fsqrt%2874%29+
cos+%28theta%29=-7%2Fsqrt%2874%29+




Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
if the angle is in quadrant 1, then all trig functions are positive.
we'll solve for the angle in quadrant 1 and then determine the angle in whatever quadrant it needs to be in, based on the information provided.

cotan(theta) = adjacent side / opposite side.

when cotan(theta) = 7/5, then adjacent side = 7 and opposite side = 5.

hypotenuse is equal to sqrt(adjacent side squared + opposite side squared) = sqrt(7^2 + 25^2) = sqrt(49 + 25) = sqrt(74)

sin(theta) = opposite side / hypotenuse = 5/sqrt(74)

cos(theta) = adjacent side / hypotenuse = 7/sqrt(74)

since cot(theta) was given as 7/5, then it must be positive and tan(theta) must also be positive because tan(theta) is the reciprocal of cot(theta)

since sec(theta) is negative, then cos(theta) must also be negative because cos(theta) is the reciprocal of sec(theta).

you have tan(theta) is positive and cos(theta) is negative.

cos(theta) is negative in quadrants 2 and 3.
tan(theta) is positive in quadrants 1 and 3.

quadrant 3 is where tan is positive and cos is negative, so the angle must be in quadrant 3.

in quadrant 3, sin is negative, therefore sin(theta) = -5/sqrt(74)

in quadrant 3, cos is also negative, therefore cos(theta) = -7/sqrt(74)

to confirm, i used my calculator to find the angle in quadrant 1.

i looked for the angle whose tan was 5/7 and whose sin was 5/sqrt(74) and whose cos was 7/sqrt(74).

that angle was 35.53767779 degrees.

i set A = to 35.53767779 degrees and solved for sin(A), cos(A), tan(A).

sin(A) = .5812381937 = 5/sqrt(74)
cos(A) = .8137334712 = 7/sqrt(74)
tan(A) = .7142857143 = 5/7

the equivalent angle in the third quadrant is equal to 35.53767779 + 180 = 215.5376778 degrees.

sin(215.5376778) = -.5812381937 = -5/sqrt(74)
cos(215.5376778) = -.8137334712 = -7/sqrt(74)
tan(215.5376778) = .7142857143 = 5/7

tan is the reciprocal of cotan, i.e. tan(theta) = 1/cot(theta).
therefore, if tan(theta) = 5/7, then cot(theta) = 7/5, as you were given.

your solution is that sin(theta = -5/sqrt(74) and cos(theta) = -7/sqrt(74) and the angle is in the third quadrant.








Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!
find sin theta and cos theta if cot theta =7/5 and sec theta <0

With cot+%28theta%29 being positive (> 0), co-function tan+%28theta%29 will also be positive (> 0). Thus, theta is either
in the 1st or 3rd quadrant. But, with sec+%28theta%29 being negative (< 0),  cos+%28theta%29 will also be negative. So,
theta will either be in the 2nd or 3rd quadrant. Therefore, theta is in the 3rd quadrant, where tan and cot 
are positive, and sec and cos are negative.

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Answers:
sin%28theta%29+=+-5%2F%28sqrt%2874%29%29+=+%28-5%2Asqrt%2874%29%29%2F%2874%29
and
cos%28theta%29+=+-7%2F%28sqrt%2874%29%29=%28-7%2Asqrt%2874%29%29%2F%2874%29
Rationalizing the denominator may be optional depending on your teacher's preference.

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Explanation

theta = theta = angle
(x,y) = coordinates of the terminal point
r = distance from origin to point (x,y)

Trig formulas to memorize (or have on a reference sheet).
sin(theta) = y/r
cos(theta) = x/r
tan(theta) = y/x		csc(theta) = r/y
sec(theta) = r/x
cot(theta) = x/y

Technically you only need to memorize 3 formulas, since each column is the reciprocal of the other.
Example: sine is the reciprocal of cosecant.

Side note: You can think of x,y,r as adjacent,opposite,hypotenuse in that order.

Because sec(theta) < 0, it must mean x < 0.
There's no way to have r < 0.

x < 0 will mean the terminal point (x,y) is to the left of the y axis.
Either it is in quadrant Q2 or Q3 (northwest and southwest quadrants respectively).

Then we also know cot(theta) = 7/5
The ratio of x over y is positive, so either x,y are both positive or both are negative.
In Q3 is when x,y are both negative.

Therefore, the terminal point must be in Q3.
The terminal point is located at (x,y) = (-7,-5)


Let's find the value of r.
x%5E2%2By%5E2=r%5E2 Due to the pythagorean theorem

r+=+sqrt%28x%5E2%2By%5E2%29

r+=+sqrt%28%28-7%29%5E2%2B%28-5%29%5E2%29

r+=+sqrt%2874%29

Then,
sin%28theta%29+=+y%2Fr

sin%28theta%29+=+-5%2F%28sqrt%2874%29%29

sin%28theta%29+=+%28-5%2Asqrt%2874%29%29%2F%28sqrt%2874%29%2Asqrt%2874%29%29

sin%28theta%29+=+%28-5%2Asqrt%2874%29%29%2F%2874%29

And also
cos%28theta%29+=+x%2Fr

cos%28theta%29+=+-7%2F%28sqrt%2874%29%29

cos%28theta%29+=+%28-7%2Asqrt%2874%29%29%2F%28sqrt%2874%29%2Asqrt%2874%29%29

cos%28theta%29+=+%28-7%2Asqrt%2874%29%29%2F%2874%29
As mentioned earlier, rationalizing the denominator may be optional.