In a right triangle, the cosine of an acute angle is defined as the ratio of the adjacent side to the hypotenuse.
cos θ =6/11 ⇔ cos θ = adjacent/hypotenuseTherefore, we know that the adjacent side of the triangle is 6 and that the hypotenuse is 11.
We can find the missing leg length by substituting b= 6 and c= 11 into the Pythagorean Theorem.
Note that when solving the equation we only considered the principal root. This is because a 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=sqrt(85)/11
-
tan θ=opp/adj
tan θ=sqrt(85)/6
-
csc θ=hyp/opp
csc θ=11/sqrt(85)
csc θ=11sqrt(85)/85
sec θ=hyp/adj
sec θ=11/6
-
cot θ=adj/opp
cot θ=6/sqrt(85)
cot θ=6sqrt(85)/85
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, 11sqrt(85) and 6sqrt(85). Let's look at how this was done for 11sqrt(85) first.
