SAT Multiple Choice Question(s)

4 cm

6 cm The figure above shows how a

rectangular piece of paper is rolled into the shape of a cylinder. If it is assumed that the 4-centimeter sides of the rectangle meet with no overlap, what is the area, in square centimeters, of the base of the cylinder?

(a)

(b)

(c)

(d)

(e)

916

9

4

4

Essential Question: How do I use trig identities to solve

equations and verify identities?

How do I use fundamental identities to verify other identities?

Reciprocal Identities

sincsc

1

sincsc

22

1

cossec

1

tancot

1

cscsin

1

seccos

1

cottan

1

Also work with powers…

See pg 454

Quotient Identities

tansin

cos

cot

cos

sin

generating the …Pythagorean Identities

(cos , sin )

cos

sin 1

a2 + b2 = c2

(cos )2 + (sin )2 = 12

cos2 + sin2 = 1

cos2 means the same thing as (cos )2

generating the…Pythagorean Identities

cos2 + sin2 = 1cos2 cos2 cos2

1 + tan2 = sec2

generating the…Pythagorean Identities

cos2 + sin2 = 1sin2 sin2 sin2

+ 1cot2 = csc2

Pythagorean Identities

cos2 + sin2 = 1

1 + tan2 = sec2

+ 1cot2 = csc2

You can also manipulate them…

These are very important!

manipulating the Pythagorean Identities

cos2 + sin2 = 1 - cos2 - cos2

sin2 = 1 - cos2

cos2 + sin2 = 1 - sin2 - sin2

cos2 = 1 - sin2

Ex. Simplify xxx cotcoscsc

Ex. Simplify xxx tancsccos

1.) Work with one side of the equation. (The complicated side first).

2.) Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

Guidelines for verifying… pg 462

3.) Look for opportunities to use the identities.

4.) If the preceding guidelines do not help, try converting all terms to sines and cosines.

Guidelines for verifying…

5.) Try something! Even making an attempt that leads to a dead end gives insight.

pg 462

factor out a GCF

Ex. Verify sin - cos2 sin =sin

= sin (1 - cos2 )

= sin (sin2 )

= sin3 Goal: Single Trig Function, if possible

Substitute w/ Pythag ID

Multiply

3

2 2 2sin sin tanx x x2 2sin (1 tan )x x

sin (sec )2 2x x

sincos

22

1x

x2

2

sin

cos

x

x

2tan x

Ex #2b Verify you try…

2tan x

2tan x

22

2

sec 1sin .

secVerify theidentity

2sin

tan

sec

2

2

22

sec

1tan

22 costan

2

2

2

coscos

sin

Ex.

sin2

2

2

sec 1

sec

2sin

2sin

Another way to do #1…

sec

sec sec

2

2 2

1

1 2 cos

sin2

Write in terms of sin or cos

cossin cos

sin

tt t

t

2cossin

sin

tt

t

sin

sin

t

t2 2sin cos

sin

t t

t

1

sin t csc t

Ex. Verify

sin(t) + cot(t) cos(t)= csc(t)

Multiply

Add, common denominator

Substitute

Substitute

1

1 sin x

1

1

sin

sin

x

x

2

1 sin

1 sin

x

x

2

1 sin

cos

x

x

2 2

1 sin

cos cos

x

x x

xx

x

x cos

1

cos

sin

cos

12

2sec tan secx x x

Ex. Verify

Multiply by the conjugate

21sec tan sec

1 sinx x x

x

What are your questions?