Question

if sec x = 13/5 and x lies in fourth quadrant, find the values of other five trigonometric functions ​

Answer 1

Answer:

1) sec X = 13/5

x lies in 4th quadrant.

cos X = 1/ secx

$\cos(x) = \frac{1}{ \frac{13}{5} } = \frac{5}{13}$

2) So, cos x = 5/13

sin X = √ 1- cos²x

$\sin(x) = \sqrt{1 - ( { \frac{5}{13} )}^{2} }$

3) sin X = -12/13

⚆As sin functions-ve in 4th quadrant.

cosec X = 1/sin X

$\csc(x) = \frac{1}{ \frac{ - 12}{13} }$

4) cosec X = -13/12

⚆As cosec functions-ve in 4th quadrant.

tan X = √ sec² x-1

$\tan(x) = \sqrt{ { \frac{13}{5} }^{2} - 1 }$

5)tan X = -12/5

⚆As tan functions-ve in 4th quadrant.

Cot x = 1/ tan X

$\cot(x) = \frac{1}{ \frac{ - 12}{5} }$

$\cot(x) = - \frac{5}{12}$

6)Cot x = -5/12

Answer 2

Therefore the values of other 5 trigonometric functions are as follows:

Sin x = 12/13; Cos x = 5/13; Tan x = 12/5; Cot x =  5/12; Cosec x = 13/12

Given:

Sec x = 13/5 and x lies in the fourth quadrant.

To Find:

The value of the other 5 trigonometric functions.

Solution:

The given question can be solved as shown below.

Concept:

1st Quadrant: Sin x, Cos x, Tan x, Cot x, Sec x, and Cosec x all are positive.

2nd Quadrant: Sin x and Cosec x are positive.

3rd Quadrant: Tan x and Cot x are positive.

4th Quadrant: Cos x and Sec x are positive.

Now coming to the question,

⇒ Sec x = ( Hypotenuse ) / ( Adjacent Side ) = 13 / 5

Hence in a right angled triangle,

Hypotenuse = 13 units

Adjacent side to angle 'x' = 5 units

Then Opposite side to 'x' = √ ( 13² - 5² )   [ ∵ By pythagoras theorem ]

⇒ Opposite side to 'x' = 12 units

Now,

⇒ Sin x = ( Opposite side ) / ( Hypotenuse ) = 12/13

⇒ Cos x = ( Adjacent side ) / ( Hypotenuse ) = 5/13

⇒ Tan x = ( Opposite side ) / ( Adjacent side ) = 12/5

⇒ Cot x = 1 / Tan x = 5/12

⇒ Sec x = 1 / Cos x = 13/5

⇒ Cosec x = 1 / Sin x = 13/12

Therefore the values of other 5 trigonometric functions are as follows:

Sin x = 12/13; Cos x = 5/13; Tan x = 12/5; Cot x =  5/12; Cosec x = 13/12

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