Question

An acute angle θ is in a right triangle with sin θ = six sevenths. What is the value of cot θ?

Answers
7 / (sqrt 13)
(sqrt 13) / 6
6 / (sqrt 13)
(sqrt 13) / 7

Answer 1

`\sin \theta =(6)/(7)\\ \cot \theta=(x)/(6)\\ x^2+6^2=7^2\\ x^2=49-36\\ x^2=13\\ x=√(13)\\ \boxed{\cot \theta =(√(13))/(6)}`

Answer 2

Consider right triangle ABC with leg BC=6, hypotenuse AB=7 and angle C such that
`\sin \angle A=(6)/(7).`

By the Pythagorean theorem, you can find the length of the second leg AC:

`AC^2+BC^2=AB^2,\\AC^2+36=49,\\AC^2=13,\\AC=√(13).`

Use the definition of
`\cot`:

`\cot \angle A=\frac{\text{adjacent leg}}{\text{opposite leg}}=(AC)/(BC) =(√(13))/(6).`

Answer:
`\cot \theta=(√(13))/(6).`