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

2 Answers
Feb 26, 2018

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

Feb 26, 2018

#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#