Transcript Document

Rigor:
You will learn how to identify, use basic
trigonometric to find trigonometric values and use
basic trigonometric identities to simplify and
rewrite trigonometric expressions.
Relevance:
You will be able to use trigonometric identities to
solve real world problems.
5-1b Trigonometric
Identities
Example 3: Find the value of each expression using the given
information.
If tan  = 1.28, find cot 𝜃
𝜋
−
2
.
𝜋
𝜋
𝑐𝑜𝑡 𝜃 −
= 𝑐𝑜𝑡 − − 𝜃
2
2
𝜋
= −𝑐𝑜𝑡
−𝜃
2
= − tan 𝜃
= −1.28
Example 4: Simplify each expression.
𝑐𝑠𝑐 𝜃 sec 𝜃 − cot 𝜃
1
1
cos 𝜃
𝑐𝑠𝑐 𝜃 sec 𝜃 − cot 𝜃 =
∙
−
sin 𝜃 cos 𝜃 sin 𝜃
1
cos 𝜃
=
−
sin 𝜃 cos 𝜃 sin 𝜃
1
cos 𝜃 cos 𝜃
=
−
∙
sin 𝜃 cos 𝜃 sin 𝜃 cos 𝜃
1
𝑐𝑜𝑠 2 𝜃
=
−
sin 𝜃 cos 𝜃 sin 𝜃 cos 𝜃
1 − 𝑐𝑜𝑠 2 𝜃
=
sin 𝜃 cos 𝜃
𝑠𝑖𝑛2 𝜃
=
sin 𝜃 cos 𝜃
sin 𝜃
cos 𝜃
= tan 𝜃
=
Example 5: Simplify each expression.
𝜋
𝑠𝑖𝑛 𝑥 cos 𝑥 − sin − 𝑥
2
2
𝑠𝑖𝑛2 𝑥 cos 𝑥 − sin
𝜋
− 𝑥 = 𝑠𝑖𝑛2 𝑥 cos 𝑥 − cos 𝑥
2
= − cos 𝑥 −𝑠𝑖𝑛2 𝑥 + 1
= − cos 𝑥 1 − 𝑠𝑖𝑛2 𝑥
= − cos 𝑥 𝑐𝑜𝑠 2 𝑥
= −𝑐𝑜𝑠 3 𝑥
Example 6: Simplify each expression.
sin 𝑥 𝑐𝑜𝑠 𝑥 1 + sin 𝑥
−
1 − sin 𝑥
cos 𝑥
=
sin 𝑥 𝑐𝑜𝑠 𝑥 cos 𝑥 1 + sin 𝑥 1 − sin 𝑥
∙
−
∙
1 − sin 𝑥 cos 𝑥
cos 𝑥
1 − sin 𝑥
sin 𝑥 𝑐𝑜𝑠 2 𝑥
1 − 𝑠𝑖𝑛2 𝑥
=
−
cos 𝑥 − sin 𝑥𝑐𝑜𝑠 𝑥 cos 𝑥 − sin 𝑥𝑐𝑜𝑠 𝑥
sin 𝑥 𝑐𝑜𝑠 2 𝑥
𝑐𝑜𝑠 2 𝑥
=
−
cos 𝑥 − sin 𝑥𝑐𝑜𝑠 𝑥 cos 𝑥 − sin 𝑥𝑐𝑜𝑠 𝑥
sin 𝑥 𝑐𝑜𝑠 2 𝑥 − 𝑐𝑜𝑠 2 𝑥
=
cos 𝑥 − sin 𝑥𝑐𝑜𝑠 𝑥
𝑐𝑜𝑠 2 𝑥 sin 𝑥 − 1
=
− cos 𝑥 sin 𝑥 − 1
cos 𝑥
−1
= −cos 𝑥
=
Example 7: Rewrite as an expression that does not involve a
fraction.
1
1 + 𝑐𝑜s 𝑥
1
1 − cos 𝑥
=
∙
1 + cos 𝑥 1 − cos 𝑥
1 − cos 𝑥
=
1 − 𝑐𝑜𝑠 2 𝑥
=
=
1 − cos 𝑥
𝑠𝑖𝑛2 𝑥
1
cos 𝑥
−
𝑠𝑖𝑛2 𝑥 𝑠𝑖𝑛2 𝑥
=
1
1 cos 𝑥
−
∙
𝑠𝑖𝑛2 𝑥 sin 𝑥 sin 𝑥
= 𝑐𝑠𝑐 2 𝑥 − cot 𝑥 csc 𝑥
−1
math!
5-1b Assignment: TX p317, 18-46 even