Given that csc θ= 265, 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 cosecant of an acute angle is defined as the ratio of the hypotenuse to the opposite side.
csc θ =26/5 ⇔ csc θ = hypotenuse/opposite
Therefore, we know that the hypotenuse of the triangle is 26 and that the opposite side to θ is 5.
We can find the missing leg length by substituting b= 5 and c= 26 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=5/26
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cos θ=adj/hyp
cos θ=sqrt(651)/26
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tan θ=opp/adj
tan θ=5/sqrt(651)
tan θ=5sqrt(651)/651
sec θ=hyp/adj
sec θ=26/sqrt(651)
sec θ=26sqrt(651)/651
cot θ=adj/opp
cot θ=sqrt(651)/5
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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, 5sqrt(651) and 26sqrt(651). Let's look at how this was done for 5sqrt(651) first.
