Circular

1
Functions
Circular
Identities
Inverses
2
Fundamental Identities

• In this lesson we will be looking at the eight
  Fundamental Identities that result from the
  definitions of the circular functions and the
  properties of the unit circle. The Fundamental
  Identities are listed in your text but you will
  need to memorize all of these.

3
Objectives

• The student will be
• able to
• State the eight Fundamental Identities

• The student will be
• able to
• Use the Fundamental Identities to verify
  identities

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An identity is an equation that is true for all
real numbers for which the equation is defined

• The equation of the unit circle is x y
  1.
• Since cos(t) x and sin(t) y on the unit
  circle, this equation would be written
• (cos(t)) (sin(t)) 1 or more briefly
• cos (t) sin (t) 1

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PYTHAGOREAN IDENTITIES

• The first of the 8 Fundamental Identities is the
• statement cos2x sin2x 1. This is the known as
  the PYTHAGOREAN IDENTITY.
• Two additional identities can be created from
• this statement through algebra strategies.

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PYTHAGOREAN IDENTITIES

• Given cos2x sin2x 1 if both sides are
• divided by cos2x, then cos2x sin2x 1 .
• cos2x cos2x cos 2x
• When simplified this becomes
• 1 tan 2 x sec 2 x.
• If both sides are divided by sin 2 x then we get
• cot 2 x 1 csc 2 x

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RATIO IDENTITIES

• Two of the Fundamental Identities are a result
• of the definitions of tangent and cotangent. By
• definition tan t y/x and cot t x/y. Since
• from the unit circle sin t y and cos t x,
• these ratios can be written as
• tan t sin t and cot t cos t
• cos t sin t

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RECIPROCAL IDENTITIES

• Finally every circular function has a function
• that is defined as it reciprocal. That is, the
• secant function is the reciprocal of the cosine
• function, cosecant is the reciprocal sine and
• cotangent is the reciprocal tangent. These are
• written
• sec t 1 csc t 1 cot t 1
• cos t sin t
  tan t

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SUMMARY
Pythagorean Identities
Ratio Identities

• cos 2 t sin 2 t 1
• tan 2 t 1 sec 2 t
• cot 2 t 1 csc 2 t

tan t sin t / cos t cot t cos t / sin t
Reciprocal Identities

• sec t 1 / cos t csc t 1 / sin t
• cot t 1 / tan t

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VERIFYING IDENTITIES

• An identity is a statement that is
• true for all elements in its domain.

To verify an identity means to show the statement
is true. This will be done using the Fundamental
Identities and algebra skills of substitution and
simplification.
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VERIFYING IDENTITIES

• SAMPLE 1
• csc t - sin t cot t cos t
• 1 - sint
• sin t
• 1 - sin2t
• sin t
• cos 2 t
• sin t

cos t cos t sin t 1 cot t cos t cot
t cos t By using a reciprocal identity, a
Pythagorean identity and algebra the statement on
the left is shown to be the same as the one on
the right.
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VERIFYING IDENTITIES

• Verifying an identity is like
• putting together a puzzle. The
• pieces are all there, but getting
• them into the correct place is
• important. Verifying an
• identity is like a proof in
• geometry. Success is
• dependent on knowing the
• basic definitions and postulates

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VERIFYING IDENTITIES

• sin t cos t cot t csc t
• sin t cos t cos t
• sin t
• sin t cos 2t
• sin t
• sin 2 t cos2t
• sin t
• 1 / sin t
• csc t csc t

SAMPLE 2
Lost? Try another one
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VERIFYING IDENTITIES

• SAMPLE 3
• cos B sec B tan B
• 1 - sin B
• 1/cosB sin B / cos B
• 1 sin B
• cos B
• 1 sin B X cos B
• cos B cos B

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SAMPLE 3 (Continued)
1 sin B X cos B cos B
cos B ( 1 sin B) cos B cos 2 B
( 1 sin B) cos B 1 - sin 2 B ( 1
sin B) cos B ( 1 sin B)(1 - sin B)
cos B cos B 1 - sin B 1 - sin B
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VERIFYING IDENTITIES
Sample 4
GOT IT ? TRY ONE MORE

• tan4tsec4t 12 sec2t
• (tan 2 t sec 2 t)( tan2 t sec 2 t)
• 1 2sec2t
• ( sec 2 t - 1 - sec 2 t)(sec 2 t 1 sec 2 t)
• 1 2sec 2 t
• (-10)(2sec 2 t- 1) 1 2sec 2 t
• 1 2sec 2 t 1 2sec 2 t