What is #cos (arcsin (5/13))#?

Question

Josiah Anderson · Accepted Answer

#12/13# First consider that : #epsilon=arcsin(5/13)# #epsilon# simply represents an angle. This means that we are looking for #color(red)cos(epsilon)!# If #epsilon=arcsin(5/13)# then, #=>sin(epsilon)=5/13# To find #cos(epsilon)# We use the identity : #cos^2(epsilon)=1-sin^2(epsilon)# #=>cos(epsilon)=sqrt(1-sin^2(epsilon)# #=>cos(epsilon)=sqrt(1-(5/13)^2)=sqrt((169-25)/169)=sqrt(144/169)=color(blue)(12/13)#

Oliver Atkinson · Answer

#12/13#

First, see #arcsin(5/13)#. This represents the ANGLE where #sin=5/13#.

That is represented by this triangle:

Now that we have the triangle that #arcsin(5/13)# is describing, we want to figure out #costheta#. The cosine will be equal to the adjacent side divided by the hypotenuse, #15#.

Use the Pythagorean Theorem to determine that the adjacent side's length is #12#, so #cos(arcsin(5/13))=12/13#.

Grace Ashcraft · Answer

undefined To find cos(arcsin(5/13)), we first use the identity arcsin(sin(x)) = x for -π/2 ≤ x ≤ π/2. Therefore, arcsin(5/13) = θ, where sin(θ) = 5/13.

Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find cos(θ). Since sin(θ) = 5/13, sin^2(θ) = (5/13)^2 = 25/169.

Substituting sin^2(θ) into the Pythagorean identity, we get:

cos^2(θ) = 1 - sin^2(θ) = 1 - 25/169 = 144/169.

Taking the square root of both sides, we find:

cos(θ) = ±√(144/169) = ±12/13.

Since θ is in the first or second quadrant (0 ≤ θ ≤ π/2), cos(θ) is positive. Therefore, cos(θ) = 12/13.

So, cos(arcsin(5/13)) = 12/13.