Identifying with Trig Identities: The Basics

Question 1

1) What do do if you shall simplify cosecant, secant, or cotangent?

2)
Solve cos * csc
______
cot

Answer 1

1)
change to sines and cosines

2) cos*(1/sin)
________ = after some steps, result = 1
cos/sin

Question 2

Proof this equation:

tan θ · csc θ = sec θ

Answer 2

Take the left side and transform in sin/cos * 1/sin, then cancel to 1/cos = sec, transform left side back, so you get sec θ = sec θ.

Question 3

Name the three Pythagorean identities for trigonomic proofs!

Answer 3

Remember 1 is hyp, cos, sin are x,y. sin² x + cos² x = 1

• > Divide every term by sin² θ to get: 1 + cot² x = csc² x
• > Divide every term by cos² θ to get: tan² x + 1 = sec² x

And their Umstellungen. So tan² x = sec² x - 1 …

Question 4

1) Which is odd and which is even. Fulfill …
sin(–x) =
cos(–x) =
tan(–x) =

2) Simplify [1 + sin(–x)][1 – sin(–x)]

Answer 4

sin(–x) = –sin x
cos(–x) = cos x = even, others are odd
tan(–x) = –tan x

2) get it to (1 – sin x)(1 + sin x), then FOIL to get 1–sin² x

Question 5

Explain and name an example for a co-function identity.

Answer 5

A co-function identity tells you that a function has the same values but is just shifted. An example is:
sin x = cos(π/2 – x). Remember, that the cos graph was just shiftet a quarter.

Question 6

What to do if you see cos(π/2 – x) or (90° – θ)

Answer 6

Replace it with sin x.

Question 7

1) What are Periodic identities?

2) In cases you solve equations and get that. How to simplify?
cos(x + 2π) =
tan(x + π) =

Answer 7

1) Graphs that are shifted with exactly one period, so they are the same just shifted by one period.

2) cos x
tan x

Question 8

What to do if you see…
sin2 + cos2
___________
cos

Answer 8

Write 1/cos.

sin2 + cos2 is Pythagoras and equals 1.

Question 9

cos t sin t
____ + ___
1 + sin t cos t Only name the first step to do!

Answer 9

Find the least common denominator: (1 + sin t) · cos t

Question 10

sin
____
sec-1 Only name the first step to do!

Answer 10

Multiply by the conjugate of the denominator. So you have to multiply by sec θ + 1 on the top and bottom of the fraction.