An acute angle θ is in a right triangle with sin θ = 6/7. What is the value of cot θ?

Trigonometry is a branch of mathematics that deals with the relation between the angles and sides of a right triangle.

Answer: In a right triangle with sin θ = 6/7, the value of cot θ = (√13) / 6.

Let's look into the stepwise solution.

Explanation:

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.