This Question 1 pt 4 of 5 0 complete Use identities to find

This Question: 1 pt 4 of 5 (0 complete) Use identities to find values of the sine and cosine functions of the function for the angle measure. 2x, given tan x =-3and cos x > 0 cos 2x = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) sin 2x= (Simplify your answer, indluding any radicals. Use integers or fractions for any numbers in the expression.)

Solution

We have given tanx=-3 and cosx>0

tanx=opposite side/adjacent side =-3/1

hypotenuse =sqrt((opposite side)²+(adjacent side)²)=sqrt(9+1)=sqrt(10)

sinx=opposite side/hypotenuse =-3/sqrt(10)

cosx=adjacent side/hypotenuse =1/sqrt(10)

cos2x=cos²x-sin²x=(1/sqrt(10))²-(-3/sqrt(10))²=1/10-9/10 =-8/10=-4/5

cos2x=-4/5

sin2x=2sinx*cosx=2*(-3/sqrt(10))*(1/sqrt(10))=-6/10=-3/5

sin2x=-3/5

We have given tan11⁰/(1-tan²11⁰)

multiply and divide by 2 above expression

tan11⁰/(1-tan²11⁰)=(2/2)*[tan11⁰/(1-tan²11⁰)]

=(1/2)*[2tan11⁰/(1-tan²11⁰)]

=1/2*tan(2*11⁰) since tan2x=2tanx/(1-tan²x)

=1/2*tan22⁰

tan11⁰/(1-tan²11⁰)=1/2*tan22⁰

We have given cos²(30⁰)-sin²(30⁰)

we know the formula cos2x=cos²x-sin²x

we take as x=30⁰

cos²(30⁰)-sin²(30⁰)=cos(2*30⁰)

=cos(60⁰)

=1/2

cos²(30⁰)-sin²(30⁰)=1/2