Question

NO LINKS!! Find the exact values of the trigonometric functions for the acute angle theta​

Answer 1

see explanation

Explanation:

given

sinΘ =
`(7)/(25)` =
`(opposite)/(hypotenuse)`

since Θ is acute then triangle is right and of the form 7- 24- 25 , that is a Pythagorean triple.

with hypotenuse = 25, adjacent = 24 and opposite = 24

Then

sinΘ =
`(opposite)/(hypotenuse)` =
`(7)/(25)`

cosΘ =
`(adjacent)/(hypotenuse)` =
`(24)/(25)`

tanΘ =
`(opposite)/(adjacent)` =
`(7)/(24)`

cscΘ =
`(1)/(sin0)` =
`(1)/((7)/(25) )` =
`(25)/(7)`

secΘ =
`(1)/(cos0)` =
`(1)/((24)/(25) )` =
`(25)/(24)`

cotΘ =
`(1)/(tan0)` =
`(1)/((7)/(24) )` =
`(24)/(7)`

Answer 2

`\sf\sin \theta=\boxed{(7)/(25)}`

`\sf \cos \theta=\boxed{(24)/(25)}`

`\sf \tan \theta=\boxed{(7)/(24)}`

`\sf\csc \theta=\boxed{(25)/(7)}`

`\sf\sec \theta=\boxed{(25)/(24)}`

`\sf\cot \theta=\boxed{(24)/(7)}`

Explanation:

Trigonometric ratios (right triangle)

`\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)`

`\sf \csc(\theta)=(H)/(O)\quad\sec(\theta)=(H)/(A)\quad\cot(\theta)=(A)/(O)`

where:

• θ is the angle.
• O is the side opposite the angle.
• A is the side adjacent the angle.
• H is the hypotenuse (the side opposite the right angle).

Given:

`\sf \sin \theta=(7)/(25)`

Therefore:

• The side opposite the angle is 7.
• The hypotenuse of the right triangle is 25.

Find the measure of the other leg of the right triangle by using Pythagoras Theorem:

`\implies \sf A^2+O^2=H^2`

`\implies \sf A^2+7^2=25^2`

`\implies \sf A^2+49=625`

`\implies \sf A^2=576`

`\implies \sf A=√(576)`

`\implies \sf A=24`

Therefore:

• O = 7
• A = 24
• H = 25

The exact values of the trigonometric functions are:

`\sf\sin \theta=(O)/(H)=\boxed{(7)/(25)}`

`\sf \cos \theta=(A)/(H)=\boxed{(24)/(25)}`

`\sf \tan \theta=(O)/(A)=\boxed{(7)/(24)}`

`\sf\csc \theta=(H)/(O)=\boxed{(25)/(7)}`

`\sf\sec \theta=(H)/(A)=\boxed{(25)/(24)}`

`\sf\cot \theta=(A)/(O)=\boxed{(24)/(7)}`