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3cos(4θ-26)+5=7,-30<= θ<45

Solution

3 cos (4 θ - 2 6^{\circ}) + 5 = 7, - 3 0^{\circ} \leq θ < 4 5^{\circ}

Solution

θ = \frac{234 0 ^{\circ} + 90 \cdot 0.84106 \dots}{360}, θ = \frac{234 0 ^{\circ} - 90 \cdot 0.84106 \dots}{360}

Radians

θ = \frac{13 π + 90 \cdot 0.84106 \dots}{360}, θ = \frac{13 π - 90 \cdot 0.84106 \dots}{360}

Solution steps

3 cos (4 θ - 2 6^{\circ}) + 5 = 7, - 3 0^{\circ} \leq θ < 4 5^{\circ}

Move

5

to the right side

3 cos (4 θ - 2 6^{\circ}) + 5 = 7

Subtract

5

from both sides

3 cos (4 θ - 2 6^{\circ}) + 5 - 5 = 7 - 5

Simplify

3 cos (4 θ - 2 6^{\circ}) = 2

3 cos (4 θ - 2 6^{\circ}) = 2

Divide both sides by

3

3 cos (4 θ - 2 6^{\circ}) = 2

Divide both sides by

3

\frac{3 cos ( 4 θ - 2 6 ^{\circ} )}{3} = \frac{2}{3}

Simplify

cos (4 θ - 2 6^{\circ}) = \frac{2}{3}

cos (4 θ - 2 6^{\circ}) = \frac{2}{3}

Apply trig inverse properties

cos (4 θ - 2 6^{\circ}) = \frac{2}{3}

General solutions for

cos (4 θ - 2 6^{\circ}) = \frac{2}{3}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

4 θ - 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n, 4 θ - 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

4 θ - 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n, 4 θ - 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

Solve

4 θ - 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n : θ = \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

4 θ - 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n

Move

2 6^{\circ}

to the right side

4 θ - 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n

Add

2 6^{\circ}

to both sides

4 θ - 2 6^{\circ} + 2 6^{\circ} = arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Simplify

4 θ = arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

4 θ = arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Divide both sides by

4

4 θ = arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Divide both sides by

4

\frac{4 θ}{4} = \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} = \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4} : \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

\frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Group like terms

= \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4} + \frac{arccos ( \frac{2}{3} )}{4}

\frac{36 0 ^{\circ} n}{4} = \frac{18 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{4}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} n}{2}

\frac{2 6 ^{\circ}}{4} = 6. 5^{\circ}

\frac{2 6 ^{\circ}}{4}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{234 0 ^{\circ}}{90 \cdot 4}

Multiply the numbers:

90 \cdot 4 = 360

= 6. 5^{\circ}

= \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

θ = \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

θ = \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

θ = \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}

Solve

4 θ - 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n : θ = 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

4 θ - 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

Move

2 6^{\circ}

to the right side

4 θ - 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

Add

2 6^{\circ}

to both sides

4 θ - 2 6^{\circ} + 2 6^{\circ} = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Simplify

4 θ = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

4 θ = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Divide both sides by

4

4 θ = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n + 2 6^{\circ}

Divide both sides by

4

\frac{4 θ}{4} = 9 0^{\circ} - \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} = 9 0^{\circ} - \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

9 0^{\circ} - \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4} : 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

9 0^{\circ} - \frac{arccos ( \frac{2}{3} )}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4}

Group like terms

= 9 0^{\circ} + \frac{36 0 ^{\circ} n}{4} + \frac{2 6 ^{\circ}}{4} - \frac{arccos ( \frac{2}{3} )}{4}

9 0^{\circ} = 9 0^{\circ}

9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

\frac{36 0 ^{\circ} n}{4} = \frac{18 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{4}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} n}{2}

\frac{2 6 ^{\circ}}{4} = 6. 5^{\circ}

\frac{2 6 ^{\circ}}{4}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{234 0 ^{\circ}}{90 \cdot 4}

Multiply the numbers:

90 \cdot 4 = 360

= 6. 5^{\circ}

= 9 0^{\circ} + \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} - \frac{arccos ( \frac{2}{3} )}{4}

Group like terms

= 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

θ = 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

θ = 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

θ = 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

θ = \frac{18 0 ^{\circ} n}{2} + 6. 5^{\circ} + \frac{arccos ( \frac{2}{3} )}{4}, θ = 9 0^{\circ} + 6. 5^{\circ} + \frac{18 0 ^{\circ} n}{2} - \frac{arccos ( \frac{2}{3} )}{4}

Solutions for the range

- 3 0^{\circ} \leq θ < 4 5^{\circ}

θ = \frac{234 0 ^{\circ} + 90 arccos ( \frac{2}{3} )}{360}, θ = \frac{234 0 ^{\circ} - 90 arccos ( \frac{2}{3} )}{360}

Show solutions in decimal form

θ = \frac{234 0 ^{\circ} + 90 \cdot 0.84106 \dots}{360}, θ = \frac{234 0 ^{\circ} - 90 \cdot 0.84106 \dots}{360}

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