http://www.ck12.org Chapter 6. Analytic Trigonometry
Solution: When doing trigonometric proofs, it is vital that you start on one side and only work with that side until
you derive what is on the other side. Sometimes it may be helpful to work from both sides and find where the two
sides meet, but this work is not considered a proof. You will have to rewrite your steps so they follow from only
one side. In this case, work with the left side and keep rewriting it until you have cos(β−π 2 ).
cot(−β)cot(π
2 −β)
sin(−β) =−cotβtanβ·−sinβ
=− 1 ·−sinβ
=sinβ
=cos(π
2 −β)
=cos(
−
(
β−π 2))
=cos(
β−π 2)
Concept Problem Revisited
The following trigonometric expression can be simplified to be equivalent to negative tangent.
[
sin(π 2 −θ)
sin(−θ)]− 1
=sinsin((π−θ)
2 −θ)
=−cossinθθ
=−tanθVocabulary
Anidentityis a mathematical sentence involving the symbol “=” that is always true for variables within the domains
of the expressions on either side. In the concept problem, the equivalent expressions are meaningless if the
denominator of the rational expression ends up as zero. This is why identities only work within a valid domain.
Cofunctionsare functions that are identical except for a reflection and horizontal shift. Examples are sine and
cosine, tangent and cotangent, secant and cosecant.
Aproofis a derivation where two sides of an expression are shown to be equivalent through a sequence of logical
steps.
Guided Practice
- Prove the quotient identity for tangent using the definition of sine, cosine and tangent.
- If cos(θ−π 2 )= 0 .68 then determine csc(−θ).
- Prove the following trigonometric identity by working with only one side.
cosxsinxtanxcotxsecxcscx= 1
Answers: - When tangent, sine and cosine are replaced with the shorthand for side ratios the equivalence becomes a matter
of algebra.