In a right triangle, the cotangent of an acute angle is defined as the ratio of the length of the adjacent side to the angle to the length of the opposite side to the angle.
Triangle:
Trigonometric Ratios:
sin θ=4sqrt(41)/41,
cos θ=5sqrt(41)/41,
tan θ=4/5,
csc θ=sqrt(41)/4,
sec θ=sqrt(41)/5
Practice makes perfect
Given that cot θ= 54, we want to sketch a right triangle with θ as the measure of one acute angle. Then, we will find the other five trigonometric ratios of θ. Let's do these things one at a time.
Drawing the Triangle
In a right triangle the cotangent of an acute angle is defined as the ratio of the length of the adjacent side to the angle to the length of the opposite side to the angle.
cot θ =5/4 ⇔ cot θ = adjacent/opposite
Therefore, we know that the length of the adjacent side to θ is 5 and that the length of the opposite side to θ is 4.
Note that when solving the equation we only considered the principal root. This is because c represents a side length and therefore must be a positive number. We can now draw the right triangle and label its three sides.
Finding Trigonometric Ratios
Having the three sides of the right triangle allows us to find the five remaining trigonometric ratios. Remember to rationalize denominators, if needed.
Function
Substitute
Simplify
sin θ=opp/hyp
sin θ=4/sqrt(41)
sin θ=4sqrt(41)/41
cos θ=adj/hyp
cos θ=5/sqrt(41)
cos θ=5sqrt(41)/41
tan θ=opp/adj
tan θ=4/5
-
csc θ=hyp/opp
csc θ=sqrt(41)/4
-
sec θ=hyp/adj
sec θ=sqrt(41)/5
-
Showing Our Work
Rationalizing Denominators
Rationalizing a denominator means eliminating any radical expression from the denominator. In the work above we needed to rationalize the denominators of two expressions, 4sqrt(41) and 5sqrt(41). Let's look at how this was done for 4sqrt(41) first.
