Question

Use the given information to find each of the following.cos θ/2 , given sin θ = - 4/5 , with 180° \u003c θ \u003c 270°

Accepted Answer

Identify the quadrant where the angle $$ 	heta $$ lies. Since $$ 180^\circ \u003c 	heta \u003c 270^\circ $$, $$ 	heta  is in $$the third quadrant, where sine is negative and cosine is also negative. Recall the half-angle formula for cosine: $$ \cos\left(\frac{	heta}{2}ight) = \pm \sqrt{\frac{1 + \cos 	heta}{2}} . We $$need to find $$ \cos 	heta $$ first. Use the Pythagorean identity $$ \sin^2 	heta + \cos^2 	heta = 1  to $$find $$ \cos 	heta . $$Given $$ \sin 	heta = -\frac{4}{5} $$, calculate $$ \cos 	heta = \pm \sqrt{1 - \sin^2 	heta} = \pm \sqrt{1 - \left(-\frac{4}{5}ight)^2} . $$Determine the correct sign of $$ \cos 	heta $$ based on the quadrant. Since $$ 	heta  is in $$the third quadrant, $$ \cos 	heta  is $$negative. Substitute the value of $$ \cos 	heta $$ into the half-angle formula and determine the sign of $$ \cos \frac{	heta}{2} $$ based on the quadrant where $$ \frac{	heta}{2} $$ lies. Since $$ 	heta  is $$between 180° and 270°, $$ \frac{	heta}{2}  is $$between 90° and 135°, which is in the second quadrant where cosine is negative.

Use the given information to find each of the following.
cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

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Key Concepts

Understanding the Quadrant and Sign of Trigonometric Functions

Half-Angle Identities

Using Pythagorean Identity to Find cos θ

Use the given information to find each of the following.cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°