An acute angle θ is in a right triangle with sin θ = 6/7. What is the value of cot θ?
Answer: In a right triangle with sin θ = 6/7, the value of cot θ = (√13) / 6.
Let's look into the stepwise solution.
Given: sin θ = 6/7
We know that for a given acute angle θ in a right triangle sin θ is expressed as:
sin θ = Opposite Side / Hypotenuse
Let's take a right-angled triangle ABC and mark the sides,
From the above diagram, we see that angle C = θ
Thus, sin θ = AB / AC [Since, AB = Opposite Side, AC = Hypotenuse]
Hence, sin θ = AB / AC = 6 / 7
We know that,
cot θ = Adjacent Side / Opposite Side = BC / AB
Thus, to calculate BC we will apply the Pythagoras theorem on triangle ABC.
According to Pythagoras theorem,
Hypotenuse2 = Base2 + Height2
From triangle ABC,
⇒ AC2 = AB2 + BC2
⇒ BC2 = AC2 - AB2
⇒ BC2 = 72 - 62 [ Since, AC = 7, AB = 6]
⇒ BC2 = 49 - 36
⇒ BC2 = 13
⇒ BC = √13
Thus, cot θ = BC / AB = (√13) / 6
We can also use Cuemath's Online Trigonometric Ratios Calculator to calculate different trigonometric ratios.
Hence, the value of cot θ = (√13) / 6.