Question

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The point given bellow is on terminal side of angle theta in standard position. Find the Exact value of each of the six trigonometric functions of theta, (9,-12) R=√x^2+√y^2

Answer 1

Given:

The point (9,-12) is on terminal side of angle theta in standard position.

To find:

The exact value of each of the six trigonometric functions of theta.

Solution:

The given point is (9,-12). Here, x-coordinate is positive and y-coordinate is negative. So, the point lies in 4th quadrant and only cos and sec are positive in 4th quadrant.

We know that,

`r=√(x^2+y^2)`

`r=√(9^2+(-12)^2)`

`r=√(81+144)`

`r=√(225)`

`r=15`

Now,

`\sin \theta=(y)/(r)=(-12)/(15)=-(4)/(5)`

`\cos \theta=(x)/(r)=(9)/(15)=(3)/(5)`

`\tan \theta=(y)/(x)=(-12)/(9)=-(4)/(3)`

`\cot \theta=(1)/(\tan \theta)=-(3)/(4)`

`\text{cosec} \theta=(1)/(\sin \theta)=-(5)/(4)`

`\sec \theta=(1)/(\cos \theta)=(5)/(3)`

Therefore, the values of six trigonometric functions of theta are
`\sin \theta=-(4)/(5),\cos \theta=(3)/(5),\tan \theta=-(4)/(3),\cot \theta=-(3)/(4),\text{cosec} \theta=-(5)/(4),\sec \theta=(5)/(3)`.