find sin theta and tan theta if cos theta = 2/3 and cot theta is >0
and
find cos theta and cot theta if sin theta = 1/3 and tan theta is <0
thanks! :)
given cosØ = 2/3 and cotØ > 0
so Ø must be in quadrant I
make a sketch of a right -angled triangle with hypotenuse 3 and adjacent 2
then y^2 + 2^2 = 3^2
y = √5
and sin Ø - √5/3
tanØ = √5/2
do the second the same way
(Ø will be in quadrant II)
For the second, draw you triangle in quadrant II
hypotenuse = 3 , opposite = 1
x^2 + 1 = 9
x = ±√8 or ±2√2 , but we are in II, so use x - -2√2
cosØ = -1/3
tanØ = - 1/2√2 ---> cotØ = -2√2
I don't know how you possibly got your answers
you have cosØ = 15/4
That would be impossible since the cosine of any angle lies between -1 and +1
I knew the angle was in the second quadrant from the CAST rule. I hope you have been taught that simple pneumonic , it tells you the sign of the 3 main trig functions in each of the 4 quadrants. Do a simple Google search for "CAST rule".
(the sine is positive in I and II,
the tangent is negative in II and IV , so where is the angle ? )
so cos would be root 15/ 4
and cot would be 1/ root 15?
one more question, how do you know it is in the second quadrant?
Sure, let's solve the first question using the given information. We know that cos(theta) = 2/3. Using the identity sin^2(theta) + cos^2(theta) = 1, we can find sin(theta).
sin^2(theta) + (2/3)^2 = 1
sin^2(theta) + 4/9 = 1
sin^2(theta) = 1 - 4/9
sin^2(theta) = 5/9
Taking the square root of both sides, we get:
sin(theta) = √(5/9)
sin(theta) = √5/√9
sin(theta) = √5/3
Now, let's find tan(theta) using the given information. We know that cot(theta) > 0. Cotangent is the reciprocal of tangent, so if cot(theta) > 0, then tan(theta) should also be > 0.
Therefore, tan(theta) > 0.
Now let's move on to the second question. We are given sin(theta) = 1/3 and tan(theta) < 0.
Using the identity sin^2(theta) + cos^2(theta) = 1, we can find cos(theta).
(1/3)^2 + cos^2(theta) = 1
1/9 + cos^2(theta) = 1
cos^2(theta) = 1 - 1/9
cos^2(theta) = 8/9
Taking the square root of both sides, we get:
cos(theta) = √(8/9)
cos(theta) = √8/√9
cos(theta) = √8/3
Now, let's find cot(theta) using the given information. We know that tan(theta) < 0. Cotangent is the reciprocal of tangent, so if tan(theta) < 0, then cot(theta) should also be < 0.
Therefore, cot(theta) < 0.
I hope this helps! Let me know if you have any more questions.
To find sin theta and tan theta, given cos theta = 2/3 and cot theta > 0, we can use the trigonometric identity of the Pythagorean theorem, which states that sin^2(theta) + cos^2(theta) = 1.
First, let's find sin theta:
1. Substitute the value of cos theta into the Pythagorean theorem identity:
sin^2(theta) + (2/3)^2 = 1.
2. Simplify the equation:
sin^2(theta) + 4/9 = 1.
3. Subtract 4/9 from both sides to isolate sin^2(theta):
sin^2(theta) = 1 - 4/9.
4. Calculate the value of sin theta:
sin^2(theta) = 5/9.
Taking the square root of both sides:
sin(theta) = sqrt(5/9).
However, remember that sin(theta) should be positive, so the correct value is: sin(theta) = sqrt(5)/3.
Next, let's find tan theta:
1. Recall that tan theta is equal to sin theta divided by cos theta:
tan(theta) = sin(theta) / cos(theta).
2. Substitute the given values into the equation:
tan(theta) = (sqrt(5)/3) / (2/3).
3. Simplify the equation by multiplying the numerator and denominator by the reciprocal of 2/3:
tan(theta) = (sqrt(5)/3) * (3/2).
4. Cancel out the common factors:
tan(theta) = sqrt(5)/2.
Therefore, sin theta = sqrt(5)/3 and tan theta = sqrt(5)/2.
To find cos theta and cot theta, given sin theta = 1/3 and tan theta < 0, we can use the trigonometric identity of the Pythagorean theorem, which states that sin^2(theta) + cos^2(theta) = 1.
First, let's find cos theta:
1. Substitute the value of sin theta into the Pythagorean theorem identity:
(1/3)^2 + cos^2(theta) = 1.
2. Simplify the equation:
1/9 + cos^2(theta) = 1.
3. Subtract 1/9 from both sides to isolate cos^2(theta):
cos^2(theta) = 1 - 1/9.
4. Calculate the value of cos theta:
cos^2(theta) = 8/9.
Taking the square root of both sides:
cos(theta) = sqrt(8/9).
However, since cos(theta) should be positive, we take the positive square root:
cos(theta) = sqrt(8)/3.
Next, let's find cot theta:
1. Recall that cot theta is equal to cos theta divided by sin theta:
cot(theta) = cos(theta) / sin(theta).
2. Substitute the given values into the equation:
cot(theta) = (sqrt(8)/3) / (1/3).
3. Simplify the equation by multiplying the numerator and denominator by the reciprocal of 1/3:
cot(theta) = (sqrt(8)/3) * (3/1).
4. Cancel out the common factors:
cot(theta) = sqrt(8).
Therefore, cos theta = sqrt(8)/3 and cot theta = sqrt(8).