Question #b2446

2 Answers
Apr 16, 2017

See below

Explanation:

#sinx=7/11#

Since #x# is in quadrant #"II"#, the angle is obtuse. This means that both #cosx# and #tanx# are negative.

#cosx=-sqrt(1-sin^2x)#

#tanx=sinx/cosx#

#cosx=-sqrt(1-(7/11)^2)=-6/11sqrt2#

#tanx=(7/11)/(-6/11sqrt2)=-7/12sqrt2#

In order to find #sin2x#, #cos2x# and #tan2x#, we need to use the double angle identities. These will be given below:

#sin2x=2sinxcosx=2(7/11)(-6/11sqrt2)=-84/121sqrt2#

#cos2x=2cos^2x-1=2(-6/11sqrt2)^2-1=23/121#

#tan2x=(2tanx)/(1-tan^2x)=(2(-7/12sqrt2))/(1-(-7/12sqrt2)^2)=-84/23sqrt2#

Apr 16, 2017

There are formulas for #sin2x, cos2x# and #tan2x# (they're called "Double-Angle Formulas")

#sin2x=2sinxcosx#
#cos2x=cos^2x-sin^2x=2cos^2x-1=1-sin^2x#
#tan2x=(2*tanx)/(1-tan^2x)#

If we are given #sinx#, we know two sides of the triangle: the hypotenuse and one leg. From there, using Pythagorean's Theorem #(a^2+b^2=c^2)#, we could find the remaining side, and thus find the ratios for both #tan# and #cos#. That will allow us to solve the Double Angle Formulas without knowing all the angles