Begin by writing [tex]$\tan (u+v)$[/tex] using [tex]$\sin (u+v)$[/tex] and [tex]$\cos (u+v)$[/tex].
[tex]$\tan (u+v)=\frac{\sin (u+v)}{\cos (u+v)}$[/tex]
Rewrite the right side using sum identities.
[tex]$\tan (u+v)=\frac{\tan (u)+\tan (v)}{1-\tan (u) \tan (v)}$[/tex]
Answer (2)
Start with the expression tan ( u + v ) = c o s ( u + v ) s i n ( u + v ) .
Apply the sum identities for sine and cosine: sin ( u + v ) = sin ( u ) cos ( v ) + cos ( u ) sin ( v ) and cos ( u + v ) = cos ( u ) cos ( v ) − sin ( u ) sin ( v ) .
Divide both the numerator and the denominator by cos ( u ) cos ( v ) .
Simplify using the identity tan ( x ) = c o s ( x ) s i n ( x ) to obtain the final formula: tan ( u + v ) = 1 − t a n ( u ) t a n ( v ) t a n ( u ) + t a n ( v ) .
Explanation
Problem Analysis We are given the initial expression for the tangent of a sum of angles in terms of sine and cosine, and the final expression in terms of tangents. Our goal is to show how to get from the first expression to the second using the sum identities for sine and cosine.
Initial Expression We start with the given expression: tan ( u + v ) = cos ( u + v ) sin ( u + v )
Applying Sum Identities We will now use the sum identities for sine and cosine: sin ( u + v ) = sin ( u ) cos ( v ) + cos ( u ) sin ( v ) cos ( u + v ) = cos ( u ) cos ( v ) − sin ( u ) sin ( v ) Substituting these into the expression for tan ( u + v ) , we get: tan ( u + v ) = cos ( u ) cos ( v ) − sin ( u ) sin ( v ) sin ( u ) cos ( v ) + cos ( u ) sin ( v )
Dividing by cos(u)cos(v) To get the right-hand side in terms of tangents, we divide both the numerator and the denominator by cos ( u ) cos ( v ) : tan ( u + v ) = c o s ( u ) c o s ( v ) c o s ( u ) c o s ( v ) − c o s ( u ) c o s ( v ) s i n ( u ) s i n ( v ) c o s ( u ) c o s ( v ) s i n ( u ) c o s ( v ) + c o s ( u ) c o s ( v ) c o s ( u ) s i n ( v )
Simplifying the Expression Now we simplify the expression. Recall that tan ( x ) = c o s ( x ) s i n ( x ) . Thus, we have: tan ( u + v ) = 1 − c o s ( u ) s i n ( u ) c o s ( v ) s i n ( v ) c o s ( u ) s i n ( u ) + c o s ( v ) s i n ( v ) tan ( u + v ) = 1 − tan ( u ) tan ( v ) tan ( u ) + tan ( v )
Final Result Therefore, we have derived the tangent addition formula: tan ( u + v ) = 1 − tan ( u ) tan ( v ) tan ( u ) + tan ( v )
Examples
The tangent addition formula is useful in physics, particularly in optics and mechanics. For example, when analyzing the trajectory of a projectile fired at an angle, or when calculating the angle of refraction of light passing through different media, this formula can simplify calculations involving angles and their tangents. It also finds applications in electrical engineering when dealing with alternating current circuits and phase angles.
We derived the sum of angles formula for tangent by applying sine and cosine sum identities. After substituting and simplifying, we arrived at tan ( u + v ) = 1 − t a n ( u ) t a n ( v ) t a n ( u ) + t a n ( v ) . This formula is useful for solving problems involving angle addition in various fields such as mathematics and engineering.
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