What are the exact values of the six trigonometric functions of the angle A in a right triangle with a = 2 and c = 11?

1 Answer
Feb 27, 2018

#sin(A)=2/11, cos(A)=sqrt(117)/11, csc(A)=11/2, sec(A)=11/sqrt(117), tan(A)=2/sqrt(117), cot(A)=sqrt(117)/2#

Explanation:

Let's draw a diagram of a right triangle with an angle #A# with the hypotenuse, #c=11# and one of the legs, #a=2.# Side #a# must be opposite to angle #A#:

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We must find #b.# Recall the Pythagorean Theorem, which states that

#a^2+b^2=c^2#

If #a=2# and #c=2,# we can substitute these values into the Pythagorean Theorem and solve for #b:#

#2^2+b^2=11^2#

#4+b^2=121#

#b^2=121-4=117#

We have #b^2#, let's take the square root of both sides to obtain #b:#

#b=sqrt(117)#

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Let's first find #sin(A)#. Recall that the sine of an angle is equal to the side opposite to the angle divided by the hypotenuse.

Therefore, #sin(A)=2/11#

Since we have #sin(A),# let's find #csc(A)#. Recall that the cosecant of an angle is equal to one over the sine of that angle, or the hypotenuse divided by the side opposite to that angle.

#csc(A)=1/sin(A)=1/(2/11)=11/2# (Because #1/(x/y)=y/x#)

Let's find #cos(A).# Recall that the cosine of an angle is equal to the side adjacent (closest to) to that angle divided by the hypotenuse. The adjacent side and the hypotenuse are NOT the same thing -- though the hypotenuse is also one of the closest sides to the angle, it is entirely distinct from the adjacent side.

#cos(A)=sqrt(117)/11#

Since we have #cos(A),# let's find #sec(A).# Recall that the secant of an angle is equal to one over the cosine of that angle, or the hypotenuse divided by the side adjacent to that angle.

#sec(A)=1/cos(A)=1/((sqrt(117)/11))=11/sqrt(117)#

Let's find #tan(A).# Recall that the tangent of an angle is equal to the side opposite to that angle divided by the side adjacent to that angle.

#tan(A)=2/sqrt(117)#

Let's finally find #cot(A).# Recall that the cotangent of an angle is equal to one over the tangent of that angle, or the side adjacent to that angle divided by the side opposite to that angle.

#cot(A)=sqrt(117)/2#