Download HW:7-5 Trig Identities

7 – 3 HW:
Trigonometric Identities
Fundamental Identities!
sin 2 θ + cos 2 θ = 1
Sum and Difference Identities
cos(A + B) = cos A cos B − sin A sin B
! sin(−θ ) = − sinθ !
tan 2 θ + 1 = sec 2 θ !
cos(−θ ) = cosθ !
cos(A − B) = cos A cos B + sin A sin B
1 + cot 2 θ = csc 2 θ !
tan(−θ) = −tan θ !
sin(A + B) = sin A cos B + sin B cos A
!
sin(A − B) = sin A cos B − sin B cos A
!
Double Angle Identities!
Half Angle Identities
sin(2A) = 2sin A cos A!
cos
1 + cos A
⎛ A⎞
=
⎝ 2⎠
2
cos(2A) = cos2 A − sin 2 A
cos(2A) = 1− 2 sin 2 A !
sin
1− cos A
⎛ A⎞
=
⎝ 2⎠
2
tan
1− cos A
⎛ A⎞
=
⎝ 2⎠
1 + cos A
cos(2A) = 2cos2 A − 1!
tan(2A) =
2tan A
!
1− tan 2 A
Techniques for proving trigonometric Identities are shown in the lecture and include the following
1. Convert everything to sine and cosine. You must show all steps and work so that it is clear
what was done on each step. I did it in my head is not an approved step.
2. Add separate fractions by getting a common numerator and denominator. Do this by
multiplying each term by a “well chosen 1” term. Add the numerators to create a useful identity or an
expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the
expression. Cross multiplication is NOT allowed. The most common way to lose points on the test
is to fail to show all steps in a clear manner.
3. Factor expressions to get common factors that will then cancel out.
4. Multiply the numerator and denominator by a “well chosen 1” term to create an identity.
Cross multiplication is NOT allowed. Use the identity to simplify the expression. The most
common way to lose points on the test is to fail to show all steps in a clear manner.
5. Use an identity to simplify the expression. Show the substitution and then show the steps to
simplify the expression. The most common way to lose points on the test is to fail to show all steps in
a clear manner.
Note: The internet has many links to examples. A web search for the topic or for you problem
can help in finding worked out solutions
Math 370 7-5 HW !
Page 1 of 2!
© 2015 Eitel
HW 7– 5!
Proving Double and Half Angle Identities. Trigonometric Identities
Use a separate sheets of paper to complete the following proofs. Put you name on the front
page and staple the pages together. Number each problem and show ALL THE STEPS.
Cross multiplication is not allowed but multiplying by a “well chosen 1 is required instead.
1− tan 2 x
1. cot x sin(2x) = 1 + cos (2x) !
2.
3. cos2 (2x) − sin 2 (2x) = cos (4 x)!
4. ( 4 cos x sin x ) 1− 2sin 2 x = sin(4 x)
4
4
5. cos x − sin x = cos(2x)!
7.
2sin x − 2sin 3 x
= sin(2x)
cos x
9.
2cos(2x)
= cot x − tan x
sin(2x)
Math 370 7-5 HW !
!
= cos(2x)
(
6.
!
1 + tan 2 x
8.
10.
2 − sec 2 x
sec 2 x
1− tan 2 x
1 + tan 2 x
)
= cos(2x)
= cos(2x)
sin 3 x + cos 3 x
= 1− sin x cos x
sin x + cos x
Page 2 of 2!
© 2015 Eitel