Pre calculus Problem of the DayHomework p. 578 1-21 odds

Simplify the following:

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a) sinπ6

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+ cosπ6

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

b) sinπ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+ cosπ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

c) sin4π3

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+ cos4π3

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

Trigonometric Identities - a statement of equality that is true for all values where the function is defined.

Reciprocal Identities:

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sinθ =1

cscθcosθ =

1secθ

tanθ =1

cotθ

€

cscθ =1

sinθsecθ =

1cosθ

cotθ =1

tanθ

Quotient Identities:

€

tanθ =sinθcosθ

cotθ =cosθsinθ

Pythagorean Identities:

€

sin2θ + cos2θ =1

€

1+ cot2θ = csc2θ

€

tan2θ +1 = sec2θ

Even/Odd Identities:

€

sin −θ( ) = −sinθ

€

cos −θ( ) = cosθ

€

tan −θ( ) = −tanθ

Verifying Trigonometric Identities

To verify a trigonometric identity we must show that one side of the identity can be simplified so that it is identical to the other side or each side can be simplified independently until they are identical.

Never treat an identity like an equation. We are not solving for the variable.

Techniques for verifying trigonometric identities.

1) Rename using the fundamental identities.

2) Rewrite a more complicated side in terms of sines and cosines.

3) Factor.

4) Combine fractional expressions using an LCD.

5) Separate a single-term quotient into two terms.

6) Multiply the numerator and denominator on one side by a binomial factor that appears on the other side of the identity.

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