Verifying Trigonometric Identities (Sec. 7.1)
Feature
Details / Examples
Trigonometric expression
An expression involving trigonometric functions. Examples:
•
Identity
To verify a trigonometric
identity, use definitions of
the functions…
and the fundamental
identities (from Sec. 6.2)
•
•
An equation that is _________ true, regardless of the values
of the variable(s).
sin θ =
cos θ =
csc θ =
sec θ =
tan θ =
cot θ =
(1) The reciprocal identities:
(2) Tangent and cotangent identities:
(3) Pythagorean identities:
How to do it: the usual
way
Transform the left-hand side of the equation into the right-hand
side, or vice-versa
Example 1
Verify the identity: sec α – cos α = sin α tan α
Transform left side =>
right, or vice-versa?
Either is okay, but it’s usually easier to turn the more
_________ expression into the ________ one.
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• Simplify: get rid of less familiar/convenient functions
Some tips
• Use fundamental identities to write sec and csc (maybe also
cot and tan) with equivalent in terms of sine and cosine.
Examples: sec(x) = 1/cos(x); tan(x) = __________
• Simplify with other methods from algebra: putting things
over common denominator, factoring out a common factor, etc.
Example 2 (you do)
Verify: sec θ = sin θ (tan θ + cot θ)
Why are trigonometric
identities important?
Applications in physics, calculus, & other topics in analytic
trigonometry, e.g., via practice with complicated trigonometric
expressions.
Example 3 (you do)
Verify for acute angle θ: tan θ = sec2 θ − 1
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But why say “acute angle”?
False “identities”
Is sec t = tan 2 t + 1 an identity? To prove it’s not, just need
one case where it fails!
Hint: consider quadrant IV.
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Identity theft
There’s no such thing (for trigonometric identities, that is).
(Note: this is a joke.)
Trigonometric substitution
Changing the form of an equation by expressing it in terms of a
trigonometric function.
Example 4
Express a 2 − x 2 in terms of a trig function of θ by
substituting x = a sin θ (for –pi/2 ≤ θ ≤ pi/2 and a>0).
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SNEAK PREVIEW of Section 7.2 (shhhhh!)
Trigonometric Equation
An equation that contains trigonometric expressions. Can it be
an identity? Yes. Does it have to be an identity? ______ What
if it’s not?
Example 5
If tan θ = 1.2738, approximate all values of θ to the nearest
0.0001 radian, for 0 ≤ θ < 2π. Hint: the tangent function has a
period of π.
Does this look familiar? Hint: see last test, problem 4 .
Example 6 (you do)
DAB, April 2011
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