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

NO LINKS!! Find the exact values of the trigonometric functions for the acute angle-example-1
User J Jiang
by
8.0k points

2 Answers

3 votes

Answer:

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)

User Mohammad Falahat
by
8.2k points
3 votes

Answer:


\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)}

NO LINKS!! Find the exact values of the trigonometric functions for the acute angle-example-1
User Uiron
by
7.6k points

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