How do you simplify the expression #sin (arctan (1/4) + arccos (3/4) )#?

Question

Zoe Austin · Accepted Answer

#sin (arctan(1/4)+arccos(3/4))=(3sqrt17+4sqrt119)/68#

Let #A=arctan(1/4)# Let #B=arccos(3/4)#

#sin (A+B)=sin A cos B + cos A sin B#

#sin (A+B)=(1/sqrt17)(3/4) + (4/sqrt17) (sqrt7/4)#

#sin (A+B)=(3+4sqrt7)/(4sqrt17)#

#sin (A+B)=(3sqrt17+4sqrt119)/68#

#sin (arctan(1/4)+arccos(3/4))=(3sqrt17+4sqrt119)/68#

God bless....I hope the explanation is useful.

Violet Anderson · Answer

undefined To simplify the expression sin (arctan (1/4) + arccos (3/4)), you can use the trigonometric identities and properties. First, find the values of arctan (1/4) and arccos (3/4).

arctan (1/4) represents the angle whose tangent is 1/4. Thus, if we take the tangent of this angle, we get 1/4. Similarly, arccos (3/4) represents the angle whose cosine is 3/4. So, if we take the cosine of this angle, we get 3/4.

Now, find the sine of the sum of these angles. You can use the sum formula for sine: sin (a + b) = sin(a)cos(b) + cos(a)sin(b).

Substitute the values of sin(arctan(1/4)), cos(arctan(1/4)), sin(arccos(3/4)), and cos(arccos(3/4)) into the formula, then calculate the expression sin(arctan(1/4) + arccos(3/4)).

Finally, simplify the expression to get the result.