Question

Determine the exact values of the trigonometric functions of the acute angle A, given tan A =3/7 ?

Answer 1

see explanation

Explanation:

Tan A = 3/7 specifies a right triangle with base = 7, height = 3.

(because by definition Tan A = height/base).

The hypotenuse of the triangle is then `sqrt(7^2 + 3^2) = 7.616` (rounding).

And with this information, you can calculate the other trig values for the triangle:

`sin a = 3/7.616`

`cos a = 7/7.616`

...and, as a final sanity check, you can use the alternate definition for Tan A:

`tan a = (sin a)/(cos a)`

`= (3/7.616)/(7/7.616)`

`= 3/7.616 * 7.616/7`

`= 3/7`

...which checks out.

I won't calculate sec a, csc a, cot a for you, you should be able to take it from here, since:

`sec a = 1/(cos a)`, `csc a = 1/(sin a)`, and `cot a = 1/(tan a)`

GOOD LUCK

Answer 2

`sin(A) = (3sqrt58)/58`
`csc(A) = sqrt58/3`
`cos(A) = (7sqrt58)/58`
`sec(A) = sqrt58/7`
`cot(A) = 7/3`

Explanation:

Given: `tan(A) = 3/7, 0< A < pi/2`

Use the identity `cot(A) = 1/tan(A)`

`cot(A) = 7/3`

Use the identity `1+tan^2(A)=sec^2(A)`

`1 + (3/7)^2= sec^2(A)`

`1 + 9/49= sec^2(A)`

`sec^2(A) = 58/49`

`sec(A) = sqrt58/7`

Using the identity `cos(A) = 1/sec(A)`

`cos(A) = (7sqrt58)/58`

Using the identity `tan(A) = sin(A)/cos(A)`

`3/7 = sin(A)/((7sqrt58)/58)`

`sin(A) = (3sqrt58)/58`

Use the identity `csc(A) = 1/sin(A)`

`csc(A) = sqrt58/3`