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)=\square[/tex]
Answer (2)
We derived the expression for tan ( u + v ) using the identities for sine and cosine. By applying the sum identities, we simplified the expression to 1 − t a n ( u ) t a n ( v ) t a n ( u ) + t a n ( v ) . This formula is useful for combining angles in trigonometric problems.
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Start with the identity 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 ) .
Substitute these into the original equation: tan ( u + v ) = c o s ( u ) c o s ( v ) − s i n ( u ) s i n ( v ) s i n ( u ) c o s ( v ) + c o s ( u ) s i n ( v ) .
Divide the numerator and denominator by cos ( u ) cos ( v ) and simplify to obtain the final result: tan ( u + v ) = 1 − t a n ( u ) t a n ( v ) t a n ( u ) + t a n ( v ) .
1 − tan ( u ) tan ( v ) tan ( u ) + tan ( v )
Explanation
Given Identity We are given the identity tan ( u + v ) = c o s ( u + v ) s i n ( u + v ) and asked to rewrite the right side using sum identities.
Using Sum Identities We need to use the sum identities for sine and cosine.
Sine Sum Identity Write the sum identity for sin ( u + v ) which is sin ( u + v ) = sin ( u ) cos ( v ) + cos ( u ) sin ( v ) .
Cosine Sum Identity Write the sum identity for cos ( u + v ) which is cos ( u + v ) = cos ( u ) cos ( v ) − sin ( u ) sin ( v ) .
Substitute Identities Substitute these identities into the expression for tan ( u + v ) to get tan ( u + v ) = cos ( u ) cos ( v ) − sin ( u ) sin ( v ) sin ( u ) cos ( v ) + cos ( u ) sin ( v ) .
Divide by cos(u)cos(v) Divide both the numerator and the denominator by cos ( u ) cos ( v ) . This is a clever trick that allows us to rewrite the expression in terms of tangents.
Rewriting the Expression This gives 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 Fractions Simplify the fractions to get 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 ) .
Final Result Rewrite using the definition of tangent to get tan ( u + v ) = 1 − tan ( u ) tan ( v ) tan ( u ) + tan ( v ) .
Examples
The tangent sum formula is useful in physics, particularly in mechanics and electromagnetism, when dealing with angles of rotation or superposition of waves. For instance, if you have two rotations defined by angles u and v , the tangent of the combined rotation angle u + v can be calculated using this formula. This is also applicable in electrical engineering when analyzing the phase angles of alternating current circuits. Understanding trigonometric identities like this allows engineers and physicists to simplify complex calculations and gain insights into system behavior.
