Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Aim: How do we use the principals of trigonometry to solve problems?

Do Now:

3If sin , the values of in

2the interval 0 360 are

1) 60 and 120 3) 120 and 300

2) 120 and 200 4) 240 and 300

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Find the value of w to the nearest tenth.

w

10 represents the side adjacent to the 540 angle.

w represents the side opposite the 540 angle.

Tangent ratio

Tan 540 = Leg opposite 540 Leg adjacent to 540

Tan 540 = w 10 1.37638192 = w

10

w = (10)1.37638192 = 13.7638192 . . .

10540

w 13.8

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

All but two of the pyramids built by the ancient Egyptians have faces inclined at 520. Suppose an archaeologist discovers the ruins of a pyramid. Most of the pyramid has eroded, but she is able to determine that the length of a side of the square base is 82m. How tall was the pyramid, assuming its faces were inclined 520?

520

h

Tan 520 = h/41

1.279941. . . = h/41

(41)1.279941. . . = h

52.4776. . . = h

52 meters h 41 m

82 m82 m

520

Height h

m.

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

1st Degree Trig Equations

algebraic trigonometric2 1 0x 2cos 1 0

what values of x make this

statement true?

conditional equalities

what values of make this

statement true?

2 1 0x 2cos 1 0 1 1 1 1

2 1x 2cos 1 1

2x

1cos

2

1 1cos ?

2 o60 or

3

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

1st Degree Trig Equations2cos 1 0

1 12cos 1

1cos

2

1 1cos 60 or

2 3

Periodic Function - period of 2

Domain = | Real numbers

Range = | 1 1

x x

y y

2

-5 5

g x = 0.5

f x = cos x QI QII QIII QIV

cosine is positive in QI and QIV

52 and 2

3 6k k

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Beware!

3sin 5 7 5

sin 4

1sin 4 ?!!!

5

3sin 12

no solution

or

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Model Problem

Find for values between 0 and 360o to the nearest degree for

3 tan - 5 = 75

tan 4

5

3tan 12

arctan 4 76 or 1.3258 radians

tan is positive in QI and QIII

76o and 256o

10

8

6

4

2

-2

-4

5

q x = 7

h x = 3tan x -5

QI QII QIII QIV

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Model Problem

Solve for cos : cos = 3 cos + 1 for all values between 0 and 2.

cos = 3 cos + 13cos 3cos

2cos 1 1

cos2

1arccos reference angle of

2 3

cosine is negative in QII and QIII

2 in QII =

3 34

in QIII = 3 3

4

3

2

1

-1

-2

2 4 6

g x = 3cos x +1

f x = cos x

QI QII QIII QIV

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Model Problem

Solve for : 5 (sin + 3) = sin + 12 for all values between 0 and 360o to the nearest minute.

3arcsin reference angle of 48.59

4

sine is negative in QIII and QIV

in QIII 180 48.59 228.59

in QIV 360 48.59 311.41

5 (sin + 3) = sin + 125sin 15 sin 12

4sin 3 3

sin4

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Model Problem

Solve for : 5 (sin + 3) = sin + 12 for all values between 0 and 360o to the nearest minute.

=228.59 228 .59 60 228 35'

=311.41 311 .41 60 311 25'

20

15

10

5

-5

2 4 6

g x = sin x +12

f x = 5sin x +3

QI QII QIII QIV

Aim: Solving Linear Trig Equations Course: Alg. 2 & Trig.

Regents Prep

If 0 < < 360o, find all values of that satisfy the equation -4 cos = 1

1. 104o, 256o

2. 76o, 104o

3. 24o, 104o, 156o, 256o

4. 76o, 104o 156o, 256o