Calculate the exact values of the cot(theta) and sin(theta) for the acute angle theta if sec(theta)= the square root of 6.
Thank you so much!
sec Ø = √6
so cosØ = 1/√6
sketch a triangle with hypotenuse √6, adjacent = 1, let the opposite be y
1^2 + y^2 = √6^2
y^2 = 5
y = √5
then sinØ = √5/√6
cotØ = 1/√5
Could you tell me why you squared those numbers? Is there a formula?
If I was doing sin instead, would 1^2 +y^2=√6^2 be in a different order?
Also how did you get the 5? Isn't the square of 6 to the second power just 6?
To find the exact values of cot(theta) and sin(theta), we need to first determine the value of theta.
Given that sec(theta) = √6, we can use the reciprocal identity of secant to find the value of cosine.
Reciprocal identity: sec(theta) = 1/cos(theta)
So, 1/cos(theta) = √6
To find cos(theta), we can take the reciprocal on both sides of the equation:
cos(theta) = 1/√6 = √6/6
Since theta is an acute angle, cosine is positive.
Now, let's find the value of sin(theta) using the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Substituting the value of cos(theta) we just found:
sin^2(theta) + (√6/6)^2 = 1
sin^2(theta) + 6/36 = 1
sin^2(theta) + 1/6 = 1
sin^2(theta) = 1 - 1/6
sin^2(theta) = 6/6 - 1/6
sin^2(theta) = 5/6
Taking the square root of both sides:
sin(theta) = ±√(5/6)
Since theta is an acute angle, sin(theta) is always positive. Hence:
sin(theta) = √5/√6
Finally, to find the exact value of cot(theta), we can use the quotient identity:
cot(theta) = cos(theta) / sin(theta)
cot(theta) = (√6/6) / (√5/√6)
Simplifying, we can multiply the numerator and denominator by the conjugate of the denominator to rationalize it:
cot(theta) = (√6/6) * (√6/√5) = (√36/6√5) = 6/√30
Therefore, the exact values are:
cot(theta) = 6/√30
sin(theta) = √5/√6