Transcript 7.6 Exploration: Trig Identities

7.6 Exploration:
Trig Identities
Honors Analysis
Learning Target: I can
develop trigonometric
identities
Identity #1:
Using either the right triangle or
coordinate definition, find a simplified
relationship for the ratio:
sin 𝑥
cos 𝑥
Identity #2:
 Evaluate
each of the following:
sin2 30° + cos 2 30°
sin2 45° + cos 2 45°
sin2 60° + cos 2 60°
sin2 90° + cos 2 90°
Conclusion: 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 =? ?
Proof of Pythagorean Identity:
2
2
𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 = 1
 Use
the triangle below to prove the
identity shown above. Hint: How can the
hypotenuse be labeled using a and b?
Deriving the
Pythagorean Identities:
 Divide the equation 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃
𝑠𝑖𝑛2 𝜃 and simplify where possible.
the result?
= 1 by
What is
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 by 𝑐𝑜𝑠 2 𝜃 and
simplify where possible. What is the result?
 Divide
Trig Identities Summary:
1.)
sin 𝑥
cos 𝑥
= tan 𝑥
2.) 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
3.) 𝑡𝑎𝑛2 𝜃 + 1 = 𝑠𝑒𝑐 2 𝜃
4.) 𝑐𝑜𝑡 2 𝜃 + 1 = 𝑐𝑠𝑐 2 𝜃
Simplify:
 sin 𝜃 (csc 𝜃
 𝑠𝑖𝑛2 𝜃
− sin 𝜃)
𝑐𝑜𝑡 2 𝜃
𝑐𝑠𝑐 2 𝜃−𝑐𝑜𝑡 2 𝜃

1−𝑐𝑜𝑠 2 𝜃
𝑐𝑜𝑠𝜃

1−sin 𝜃
−
 −5𝑠𝑖𝑛3 𝑥
cos 𝜃
1+sin 𝜃
− 5 sin 𝑥 𝑐𝑜𝑠 2 𝑥
Strategies for simplifying
trigonometric expressions
 Write
trig functions in term of sin/cos/tan
 Look for Pythagorean identities (may
need to factor out GCF to find)
 Fractions: Find a common denominator &
combine
 Fractions: Break up a sum/difference in
the numerator as two fractions with the
same denominator
Verify the trig identity
sin 𝑥 + csc 𝑥 = 1
Verify the trig identity:
1
1
+
=1
2
2
𝑠𝑒𝑐 𝜃 𝑐𝑠𝑐 𝜃
Verify the trig identity:
cos 𝑥
tan 𝑥 =
sin 𝑥 𝑐𝑜𝑡 2 𝑥
Verify the trig identity:
2
1 + 𝑡𝑎𝑛 𝜃
2
=
𝑡𝑎𝑛
𝜃
𝑐𝑠𝑐 2 𝜃
Trig Identity Tips
 Make
obvious replacements
 Convert reciprocal functions to
sin/cos/tan where possible
 If factored, multiply out
 If not factored, factor out GCF
 Try converting all values to sin/cos form
 If there is a sum or difference in the
numerator, split it up into two fractions
(CAREFUL!! You can’t split up denom!)
 Sum/difference of fractions – find
common denominator and add
Multiple Angle Identities
Sum/Difference
sin 𝛼 + 𝛽 = sin 𝛼 𝑐𝑜𝑠𝛽 + sin 𝛽 cos 𝛼
sin 𝛼 − 𝛽 = sin 𝛼 𝑐𝑜𝑠𝛽 − sin 𝛽 cos 𝛼
sin 𝛼 ± 𝛽 = sin 𝛼 𝑐𝑜𝑠𝛽 ± sin 𝛽 cos 𝛼
cos 𝛼 + 𝛽 = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽
cos 𝛼 + 𝛽 = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽
cos 𝛼 ± 𝛽 = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽
How could you find tangent of a sum??
Multiple Angle Identities
 sin 2𝜃
= 2 sin 𝜃 cos 𝜃
= 𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃
 cos 2𝜃 = 1 − 2 𝑠𝑖𝑛2 𝜃
 cos 2𝜃 = 2𝑐𝑜𝑠 2 𝜃 − 1
 cos 2𝜃
Simplify
 cos
 sin
2
20 cos 40 – sin 20 cos 40
10 cos 40 + sin 40 cos 10
sin 50 cos 50
 1 − 2 𝑠𝑖𝑛2 40
Evaluate:
 sin 75°
 cos 15°
 cos 105°
3
5
3
,
5
 If
sin 𝛼 = , find sin 2𝛼.
 If
sin 𝛼 =
find cos 2𝛼.
Ch. 7 Test Review: Identities
Simplify:
cos 𝑥 − cos 𝑥 𝑠𝑖𝑛2 𝑥
Simplify:
sec 𝑥 + cos 𝑥
sec 𝑥
Simplify:
sin 𝑥 cot 𝑥
Simplify:
Simplify:
Simplify:
Simplify:
Simplify: