if sinθ=-4/5 and θ is in quadrant III, determine the exact value(s) of cos(θ - pi/6)

Answer 1

cos(x-30)=? when sinx=-4/5

cosx =-3/5

cos(x-30) = cosxcos30+sinxsin30 = (sqr3/2)cosx +.5sinx

cos(x-30) =-.3sqr3 -.4 exactly

check the answer

sinx =-4/5, x =sin^-1(-4/5) = -53.13 or 180=53.13 = 233.13 degrees

and cos(233.13-30)=cos(203.13)= about-.92

-.3(1.732)-.4 = about-.5196-.4 = about -.92

Answer 2

If θ is in Quadrant III, then both the cosine and the sine are negative.

The sine is y / hypotenuse and cosine is x / hypotenuse.

Since sin(θ) = -4/5, cos(θ) = -3/5. Why? If y = -4 and hypotenuse = 5, then we are dealing with with a 3-4-5 right triangle. So x must equal -3 ; remember that θ is in the third quadrant where all x's and y's are negative.

Recall the cosine difference formula:

cos ⁡ ( α − β ) = cos ⁡ α ⋅ cos ⁡ β + sin ⁡ α ⋅ sin ⁡ β 

cos(θ - π/6)) = cos(θ)*cos(π/6) + sin(θ)*sin(π/6)

= (-3/5)*(0.5√3) + (-4/5)*(0.5)

= -0.3√3 - 0.4

= -0.4 - 0.3√3