Question

Work each problem.Find the exact values of sin x, cos x, and tan x, for x = π/12 , usinga. difference identitiesb. half-angle identities.

Accepted Answer

Identify the angle $$ x = \frac{\pi}{12} $$ radians, which is equivalent to 15 degrees. Recognize that $$ \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} $$, so you can use difference identities with angles $$ \frac{\pi}{3} $$ and $$ \frac{\pi}{4} . $$For part (a), apply the difference identities for sine, cosine, and tangent:  
- $$ \sin(a - b) = \sin a \cos b - \cos a \sin b $$  
- $$ \cos(a - b) = \cos a \cos b + \sin a \sin b $$  
- $$ 	an(a - b) = \frac{	an a - 	an b}{1 + 	an a 	an b} $$  
Use $$ a = \frac{\pi}{3} $$ and $$ b = \frac{\pi}{4} $$ and substitute the exact values of sine, cosine, and tangent for these standard angles. For part (b), use the half-angle identities to find $$ \sin \frac{	heta}{2} $$, $$ \cos \frac{	heta}{2} $$, and $$ 	an \frac{	heta}{2} $$ where $$ 	heta = \frac{\pi}{6} $$ because $$ \frac{\pi}{12} = \frac{1}{2} 	imes \frac{\pi}{6} . $$The half-angle formulas are:  
- $$ \sin \frac{	heta}{2} = \pm \sqrt{\frac{1 - \cos 	heta}{2}} $$  
- $$ \cos \frac{	heta}{2} = \pm \sqrt{\frac{1 + \cos 	heta}{2}} $$  
- $$ 	an \frac{	heta}{2} = \pm \sqrt{\frac{1 - \cos 	heta}{1 + \cos 	heta}}  or$$ $$ 	an \frac{	heta}{2} = \frac{\sin 	heta}{1 + \cos 	heta} $$ Determine the correct signs for the half-angle values based on the quadrant where $$ x = \frac{\pi}{12} $$ lies (first quadrant, so sine, cosine, and tangent are positive). Substitute the exact value of $$ \cos \frac{\pi}{6} $$ into the half-angle formulas. Simplify the expressions under the square roots and the fractions to get the exact values of $$ \sin \frac{\pi}{12} $$, $$ \cos \frac{\pi}{12} $$, and $$ 	an \frac{\pi}{12} $$ using both methods, without calculating decimal approximations.

Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities.

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Concetti chiave

Difference Identities

Half-Angle Identities

Exact Values of Common Angles