11.1 - Basic Trigonometry Identities

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WARM-UP
Prove sin2 x cos2 x 1
This is one of 3 Pythagorean Identities that we
will be using in Ch. 11. The other 2 are
2
11.1 - Basic Trigonometry Identities

• Objective to be able to verify basic trig
  identities

You must know and memorize the following.
Pythagorean Identities
Tangent/Cotangent Identities
sin2 x cos2 x 1
1 tan2 x sec2 x
1 cot2 x csc2 x
Reciprocal Identities
Cofunction Identities
sin2 x (sin x)2
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Summary ofDouble-Angle Formulas
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All Students Take Calculus.
Quad I

• Quad II

cos(A)gt0 sin(A)gt0 tan(A)gt0 sec(A)gt0 csc(A)gt0 cot(A
)gt0
cos(A)lt0 sin(A)gt0 tan(A)lt0 sec(A)lt0 csc(A)gt0 cot(A
)lt0
cos(A)lt0 sin(A)lt0 tan(A)gt0 sec(A)lt0 csc(A)lt0 cot(A
)gt0
cos(A)gt0 sin(A)lt0 tan(A)lt0 sec(A)gt0 csc(A)lt0 cot(A
)lt0
Quad IV
Quad III
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Reference Angles
Quad I
Quad II
? ?
? 180 ?
? p ?
? ? 180
? 360 ?
? 2p ?
? ? p
Quad III
Quad IV
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We can prove the trigonometric identities for
specific angles.
Ex1) 1 tan2 45 ? sec2 45
Ex2) (sin 30)( sec 30)(cot 30) ? 1
We can prove the trigonometric identities by
using the trigonometric ratios.
Ex4) (sin x) (csc x) ? 1
Ex3) (tan x) (cos x) ? sin x
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Prove each using the trigonometric identities.
Ex6) (1 cos x)(1 cos x) ? sin2 x Ex7) 1
? csc2 x Ex8) Ex9)
Can you prove trig identities for specific
angles? Using trig ratios? Or, using trig
identities? Assignment ws11.1
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11.2a Trigonometric Identities
Objective To use trigonometric identities and
factoring to do basic trig proofs.

• Helpful Hints
• Factor and cancel
• Start with the more complicated side and
  manipulate it to equal the other side.
• Convert to sines and cosines.
• Do you need a common denominator?
• YOU MAY NOT CROSS THE ARROW!!!!

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Prove each identity.
Ex5) csc x ? sin x (cos x)(cot x)
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Write each in terms of sine. (What does this
mean?)
Write each in terms of cosine. (What does this
mean?)
Can you use the trigonometric identities to work
a trig proof? Assign WS 11.2a
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11.2a Solutions
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11.2b More Trigonometric Identities
Objective To continue trigonometric proofs
using trig identities.
Ex2) (cot2 ? )(sec2 ?) 1 cot2 q
Ex3) cos x(csc x tan x) ? cot x sin x
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Have you memorized your trig identities? Are you
ready for an IDENTITY QUIZ? Assignment Worksheet
11.2b
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WARM-UP

1. Given a triangle with a5, b7, and c9. Find
   all of its angles.
2. Given a triangle with A60, c12, and b42. Find
   the remaining side and angles.

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WARM-UP

• The expressions sin (A B) and cos (A B)
  occur frequently enough in math that it is
  necessary to find expressions equivalent to them
  that involve sines and cosines of single angles.
  So.
• Does sin (A B) Sin A Sin B
• Try letting ?A 30? and ?B 60?

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11.3 Sum and Difference Formulas
Objective To use the sum and difference
formulas for sine and cosine.
sin (? ?) sin a cos b sin b cos a sin (? -
?) sin a cos b - sin b cos a
1. This can be used to find the sin 105?.
HOW? 2. Calculate the exact value of sin 375?.
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cos (? ?) cos a cos b - sin a sin b cos (? -
?) cos a cos b sin a sin b
Note the similarities and differences to the sine
properties.
3. This can be used to find the cos 285?.
HOW? 4. Calculate the exact value of cos 345?.
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Write each expression as the sine or cosine of a
single angle.
cos 80? cos 20? sin80? sin 20? ? sin 30? cos
15? sin15? cos30? ? cos 12? cos x? - sin12?
sin x? ?
Do you understand the difference between the sum
and difference properties for sine and cosine
difference? Assignment ws 11.3
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11.5a - Solving Trigonometric Equations
Objective To solve trigonometric equations
involving special angles.
What does it meant to solve over 0? lt x lt 360? ?
What does it meant to solve over 0 lt x lt 2p ?
Recall You need the values of your special
angles. ?Do you have your unit circle? ?Can
you reproduce your special triangles? ?Do you
remember how to determine the values of your
axis angles?
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Solve over the interval 0? lt x lt 360?.
Solve over the interval 0 lt x lt 2p.
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Just a few more!!! Solve these over the interval
0? lt x lt 360? .What happens when the angle
doesnt x????
Can you solve trig equations? Do you
know/remember how to pick the appropriate
quadrant for each answer? Assign Worksheet 11.5a
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11.5b More Equations
Objective To solve trigonometric equations that
do not have special angle answers.
These are similar to the problems from 11.5a,
except you will need your calculator to solve
these. You will also need to know how to find
angles in each of the four quadrants.
Ex1 5 cos2 x 15 cos x 3 0 Ex2 49sin2 x
1 0
Ex3 sin 3x sec x 3 sin 3x Ex4 4csc2 x
8cscx 5
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Try this one!Ex5 2cos2 x 4 cos x 1 0
Just so you dont forget!Ex6 sin 4x ½
Assign WS 11.5b And. Start studying for your Ch
11 test! Look over your proof quiz too!
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Chapter 11 Review
What have we covered?

• Proving identities using specific angles,
  trigonometric ratios and trigonometric
  identities. (Basically the first quiz)
• Trigonometric Identities (see note packet)
• Sum and difference properties for sine and
  cosine.
• Solving trigonometric equations. You will have a
  unit circle for this test.
• How do you know what quadrant you should choose
  for your answers? How do you determine answers
  for angles other than x? (sin 2x 1)
• This is the last test! ?

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11.1 - Basic Trigonometry Identities

• Objective to be able to verify basic trig
  identities

You must know and memorize the following.
Pythagorean Identities
Tangent/Cotangent Identities
sin2 x cos2 x 1
1 tan2 x sec2 x
1 cot2 x csc2 x
Reciprocal Identities
Cofunction Identities
sin2 x (sin x)2
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