Question

if sec theta =5/4 then sin theta <0 then cot theta is equivalent

Answer 1

Question :

sec θ = 5/4 .

Find cos θ .

Answer:

$\huge\boxed{\boxed{\green{\dfrac{4}{3}}}}$

Step-by-step explanation:

There are two methods to solve this .

First method involves the use of the formula of secθ and cot θ .

sec θ = longest side / side adjacent to angle

In simple words :

sec θ =  hypotenuse / base  .

Given :

⇒ hypotenuse / base = 5/4

Let the hypotenuse be 5x .

Then the base is 4x .

By Pythagoras theorem :

( Height )² + ( Base )² = ( Hypotenuse )²

⇒ ( H )² + ( 4x )² = ( 5x )²

⇒ H² + 16 x² = 25 x²

⇒ H² = 25 x² - 16 x²

⇒ H² = 9 x²

⇒ H = 3 x

The height is found to be 3 x .

cot θ = side adjacent to angle / side opposite side

⇒ cot θ = 4x / 3x

⇒ cot θ = 4/3

∴ cot θ = 4/3 .

Another method involves the use of formulas relating to trigonometric equations .

Formulas used :-

$\bigstar sec\theta =\dfrac{1}{cos\theta}\\\\\bigstar cot\theta=\dfrac{cos\theta}{sin\theta}\\\\\bigstar sin^2\theta=1-cos^2\theta\\\\\bigstar sin\theta=\sqrt{1-cos^2\theta}$

Calculations :-

$cot\theta=\dfrac{cos\theta}{sin\theta}\\\\\implies cot\theta=\dfrac{cos\theta}{\sqrt{1-cos^2\theta}}\\\\\implies cot\theta=\dfrac{\dfrac{1}{sec\theta}}{\sqrt{1-\frac{1}{sec^2\theta}}}\\\\\\\textbf{Put the values of sec\theta }\\\\\implies cot\theta=\dfrac{\dfrac{1}{sec\theta}}{\sqrt{\dfrac{sec^2\theta-1}{sec^2\theta}}}\\\\\implies cot\theta=\dfrac{1}{\sqrt{sec^2\theta-1}}\\\\\implies cot\theta=\dfrac{1}{\sqrt{\dfrac{25}{16}-1}}\\\\\implies cot\theta=\dfrac{1}{\sqrt{\dfrac{9}{16}}}$

$\implies cot\theta=\dfrac{1}{\sqrt{\dfrac{9}{16}}}\\\\\implies cot\theta=\dfrac{1}{\dfrac{3}{4}}\\\\\implies cot\theta=\dfrac{4}{3}$

Answer 2

Solution :  

Given -  

➨ Hypotenuse / base = 5/4  

Let the Hypotenuse and base respectively be 5y and 4y.  

By Pythagoras theorem :-  

➨ Height² + Base² = Hypotenuse²  

➨ H² + 4y² = 5y²  

➨ H² + 16y² = 25y²  

➨ H² = 25 y² - 16 y²  

➨ H² = 9 y²  

➨ H = 3 y  

Now , the Height = 3 y.  

We know that ,  

Cot ∅ = Side of adjacent /side opposite side  

➨ Cot ∅ = 4y/3y  

➨ Cot ∅ = 4/3  

Hence ,  

➨ Cot ∅ = 4/3