Given that sin θ= 720, 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 sine of an acute angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
sin θ =7/20 ⇔ sin θ = opposite/hypotenuse
Therefore, we know that the length of the opposite side to θ is 7 and that the length of the hypotenuse is 20.
We can find the missing leg length by substituting b= 7 and c= 20 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
cos θ=adj/hyp
cos θ=3sqrt(39)/20
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tan θ=opp/adj
tan θ=7/3sqrt(39)
tan θ=7sqrt(39)/117
csc θ=hyp/opp
csc θ=20/7
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sec θ=hyp/adj
sec θ=20/3sqrt(39)
sec θ=20sqrt(39)/117
cot θ=adj/opp
cot θ=3sqrt(39)/7
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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, 73sqrt(39) and 203sqrt(39). Let's look at how this was done for 73sqrt(39) first.
