if sin θ = -7/13 and cos θ= 12/13, find tan θ and cot θ using Quotient Identities.
tan = sin / cos
cot = 1 / tan = cos / sin
Well, well, well, it seems we have a trigonometry problem on our hands! Don't worry, my friend, I'm here to make it as fun as possible. Let's tackle it together!
First, let's recall the definition of the tangent function. Tangent, denoted as tan θ, is defined as the ratio of the sine of an angle to the cosine of the same angle. In other words, tan θ = sin θ / cos θ.
Now that we have sin θ = -7/13 and cos θ = 12/13, we can substitute these values into our formula.
tan θ = sin θ / cos θ
= (-7/13) / (12/13)
= -7/13 * 13/12
= -7/12.
Voila! We have found the tangent of θ. But wait, there's more!
Cotangent, denoted as cot θ, is simply the reciprocal of tangent. In mathematical terms, cot θ = 1 / tan θ.
So, to find cot θ, we just need to take the reciprocal of -7/12.
cot θ = 1 / tan θ
= 1 / (-7/12)
= -12/7.
And there you have it! The tangent of θ is -7/12, and the cotangent of θ is -12/7. I hope I was able to add a touch of humor to your trigonometry adventure!
To find the values of tan θ and cot θ using the Quotient Identities, we can make use of the given values of sin θ and cos θ.
The Quotient Identities are as follows:
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Given that sin θ = -7/13 and cos θ = 12/13, we can substitute these values into the Quotient Identities.
tan θ = (-7/13) / (12/13)
= (-7/13) * (13/12) (dividing by a fraction is the same as multiplying by its reciprocal)
= -7/12
cot θ = (12/13) / (-7/13)
= (12/13) * (-13/7) (dividing by a fraction is the same as multiplying by its reciprocal)
= -12/7
Therefore, tan θ = -7/12, and cot θ = -12/7.
To find tan θ and cot θ using the Quotient Identities, we will use the formulas:
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Given that sin θ = -7/13 and cos θ = 12/13, we can substitute these values into the formulas:
tan θ = (-7/13) / (12/13)
cot θ = (12/13) / (-7/13)
Simplifying these expressions, we have:
tan θ = -7/12
cot θ = 12/-7
Therefore, tan θ = -7/12 and cot θ = -12/7.